A Graphical Approach to Precalculus with Limits (7th Edition)

Chapter 8

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General equation of parabola symmetric to x axis ... more

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; General equation of a circle centred at having... more

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; General equation of a circle centred at having... more

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; General equation of a circle centred at having... more

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; General equation of a circle centred at having... more

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; General equation of a circle centred at having... more

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; General equation of a circle centred at having... more

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; General equation of a circle centred at having... more

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; General equation of a circle centred at having... more

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Circle with centre at (3,3) and radius 0. ; ... more

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No graph possible. ; General equation of a circle ... more

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; Midpoints of two coordinates (a,b) and (c,d) is... more

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; Midpoints of two coordinates (a,b) and (c,d) is... more

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; Midpoints of two coordinates (a,b) and (c,d) is... more

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; Midpoints of two coordinates (a,b) and (c,d) is... more

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; Midpoints of two coordinates (a,b) and (c,d) is... more

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; Midpoints of two coordinates (a,b) and (c,d) is... more

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 By the definition of circle all the points of the... more

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Let and be the co-ordinates of diameter . Center... more

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Domain is and range is . ; The standard center-... more

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Domain is and range is ; The standard center-... more

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No solution ; The center-radius form of equation... more

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Domain is and range is ; The standard center-... more

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No real circle possible . ; Subtract 2 from both ... more

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Domain is and range is . ; Subtract from both ... more

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Domain is and range is ; Subtract from both side... more

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Yes, circle exist.Center is  and radius is 4 ; The... more

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Yes,the circle exist with center at and radius 3... more

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Yes, the circle exist with center at and radius 6... more

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Yes; ; 6 ; Compare the equation with the general ... more

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Yes; ; 3 ; Compare the equation with the general ... more

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No ; Compare the equation with the general form ... more

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No ; Compare the equation with the general form ... more

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Yes; ; ; Compare the equation with the general ... more

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Yes; ; ; Compare the equation with the general ... more

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Yes; ; ; Compare the equation with the general ... more

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Yes; ; ; Compare the equation with the general ... more

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Convert the equation into general form . The ... more

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; ; y-axis ; The equation is in the form of which ... more

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; ; y-axis ; The equation is in the form of which ... more

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The focus and directrix are and respectively. ... more

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The focus and directrix are and respectively. ... more

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The focus and directrix are and respectively. ... more

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The focus and directrix are and respectively. ... more

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The focus and directrix are and respectively. ... more

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; The focus and vertex are given as and ... more

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; The focus and vertex are given as and ... more

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; The focus and vertex are given as and ... more

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; Form of parabola which opens on right will be... more

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; Form of parabola which opens on left is  ; As... more

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; Focus and Vertex are given as and ... more

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; Here Focus is on right side of directrix, so ... more

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  ; Given that Focus and the directrix .The ... more

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; Given that Focus and the directrix .The ... more

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; Given that vertex and passing through  . Then ... more

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. ; Given that vertex and passing through  . Then... more

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Vertex: , Axis: , Domain: and Range: . ; Consider... more

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Vertex: , Axis: , Domain: and Range: . ; Consider... more

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Vertex: , Axis: , Domain: and Range: . ; Consider... more

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Vertex: , Axis: , Domain: and Range: . ; Consider... more

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Vertex: , Axis: , Domain: and Range: . ; Consider... more

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Vertex: , Axis: , Domain: and Range: . ; Consider... more

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Vertex: , Axis: , Domain: and Range: . ; Consider... more

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Vertex: , Axis: , Domain: and Range: . ; Consider... more

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Vertex: , Axis: , Domain: ) and Range:  . ; ... more

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Vertex: , Axis: , Domain: ) and Range:  . ; ... more

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Vertex: , Axis: , Domain: ) and Range:  . ; ... more

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Vertex: , Axis: , Domain: and Range:  . ; ... more

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Vertex: , Axis: , Domain: ) and Range:  . ; ... more

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Vertex: Axis: Domain: Range: ; To draw the graph ... more

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Vertex: Axis: Domain: Range:   ; To draw the graph... more

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Vertex: Axis: Domain: Range: ; To draw the graph ... more

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Vertex: Axis: Domain: Range: ; To draw the graph ... more

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Vertex: Axis: Domain: Range: ; To draw the graph ... more

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Vertex: Axis: Domain: Range: ; To draw the graph ... more

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Vertex: Axis: Domain: Range:   ; To draw the graph... more

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Vertex: Axis: Domain: Range:   ; To draw the graph... more

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; Find the equation of the parabola in the  form... more

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; Given where .Substitute the values in the ... more

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; Let the equation of the downward parabola be... more

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; The value of for Earth is 32.2. Thus the ... more

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; Let the equation of the parabola be:  ; Given... more

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Let be a point on the parabola. Thus the point ... more

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Mirror about the axis: and Mirror about the point... more

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; The general equation of ellipse  with center at... more

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Domain: [-3,3]; Range: [-2,2]; Foci: and ; The ... more

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Domain: [-4,4]; Range: [-6,6]; Foci: and ; The ... more

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Domain: ; Range: [-3,3]; Foci: and ; The domain ... more

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Domain: ; Range: ; Foci: and ; The domain is ... more

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Domain: ; Range: ; Foci: and ; The domain is ... more

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Domain: ; Range: ; Foci: and ; The domain is ... more

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Domain: ; Range: ; Center: ; The domain is given ... more

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; Calculate the value of foci using the ... more

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; Calculate the equation of ellipse:The ... more

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; Calculate the value of using the expression:... more

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; Calculate the minor axis:This can be computed ... more

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  ; Calculate the major axis:This can be computed ... more

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; Calculate the minor axis:This can be computed ... more

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  ; Calculate the minor axis: This can be computed... more

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; Calculate the minor axis. The minor axis can be... more

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,   and ; Calculate the equation of ellipse by ... more

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 , and ; Calculate the equation of ellipse by ... more

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, and ; Calculate the equation of ellipse by ... more

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 , and ; Calculate the equation of ellipse by ... more

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, and   ; Calculate the equation of ellipse by ... more

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Expand the expression:The expression can be ... more

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Expand the expression:The expression can be ... more

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Standard form of a hyperboal equation is , where... more

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Standard form of a hyperboal equation is , where... more

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Standard form of a hyperboal equation is , where... more

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Standard form of a hyperboal equation is , where... more

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Standard form of a hyperboal equation is , where... more

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Standard form of a hyperboal equation is , where... more

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Standard form of a hyperbola equation is , where... more

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Standard form of a hyperbola equation is , where... more

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Standard form of a hyperbola equation is , where... more

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Standard form of a hyperbola equation is , where... more

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Standard form of a hyperbola equation is , where... more

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Standard form of a hyperbola equation is , where... more

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Standard form of a hyperbola equation is , where... more

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Graph of is symmetrical about line . Hence the ... more

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; Standard form of a hyperboal equation is , ... more

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; Standard form of a hyperboal equation is , ... more

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; Standard form of a hyperboal equation is , ... more

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; Standard form of a hyperboal equation is , ... more

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; Standard form of a hyperboal equation is , ... more

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; Standard form of a hyperboal equation is , ... more

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; Given vertices are . Hence, the value for will... more

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F1 is and F2 is ; Given, the hyperbola has ... more

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; Given, vertices at so and asymtotes : . ... more

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; Given, foci at so and asymptotes . Equating ... more

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; Given vertices at and and asymptotes : . For ... more

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; Given vertices as and so it is a horizontal ... more

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; Given,1. Centre so and -22. Focus so =4 and... more

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Equation of the hyperbola is with centre at (1, 1... more

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Equation of the hyperbola is with centre at (1, 1... more

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F1 and F2 are and ; simplifying the given ... more

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The  is  = 4. ; The equation of the hyperbola is... more

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Ellipse is defined as set of all points in a plane... more

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= 1 ; The stone is placed at 12 units from the ... more

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The approximate distance between the foci is 347 ... more

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The perimeter is approximately 1789 feet. ; The ... more

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The height of the tallest truck is approximately ... more

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The point is on hyperbola. ; The equation of the ... more

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The point on hyperbola satisfies the equation  = 1... more

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The equation of the ellipse is  = 1 ; The foci are... more

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The equation of the hyperbola is  = 1 ; The foci ... more

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A set of points in plane that are equidistant from... more

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The standard form of the equation is - = ... more

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Now input the function directly in the calculator... more

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Now input the function directly in the calculator... more

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If the vertices and foci of the conic section are... more

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; The conic section can be identified as the ... more

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; The conic section can be identified as ellipse ... more

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; The conic section can be identified as ... more

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Circle ; For the equation , the coefficient of ... more

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Circle ; For the equation , the coefficient of ... more

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parabola ; For the equation , one of the variable ... more

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parabola ; For the equation , one of the variable ... more

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parabola ; For the equation , one of the variable ... more

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ellipse ; For the equation , the coefficient of ... more

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not a conic section ; For the equation , the ... more

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hyperbola ; For the equation , has the ... more

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hyperbola ; Simplify the equation by writing the ... more

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Ellipse having center at  major axis is along y-... more

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hyperbola ; Simplify the equation by writing the ... more

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parabola ; For the equation , one of the ... more

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Ellipse ; First convert the equation into ... more

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Hyperbola ; First convert the equation into ... more

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Circle ; First convert the equation   into ... more

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Circle ; First convert the equation into standard... more

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Parabola ; In the equation , the term is present ... more

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Parabola ; In the equation , the term is present ... more

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Point ; In the equation , the coefficients of and... more

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Parabola ; In the equation , the term is present ... more

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Circle ; In the equation , the coefficients of ... more

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Parabola ; In the equation , the term is present ... more

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Empty ; In the equation , the coefficients of and... more

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Empty ; In the equation , the coefficients of and... more

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Parabola and equation of parabola is ; In the ... more

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Circle ; In the equation , the coefficients of ... more

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Hyperbola ; In the equation , the term has ... more

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Ellipse ; In the equation , the coefficients of ... more

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; empty ; In the equation , the coefficients of ... more

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; empty ; In the equation , the coefficients of ... more

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Parabola ; In the equation , the term is present ... more

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; empty ; In the equation , the coefficients of ... more

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Circle. ; The circle equation is formulated as , ... more

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hyperbola ; The hyperbola equation is formulated ... more

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ellipse ; The ellipse equation is formulated as... more

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Circle. ; The circle equation is formulated as , ... more

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; The eccentricity of the ellipse is determined ... more

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3 ; The eccentricity of the hyperbola is ... more

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; The eccentricity of the hyperbola is determined... more

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; The eccentricity of the ellipse is determined ... more

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; The eccentricity of the ellipse is determined ... more

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; The eccentricity of the hyperbola is determined... more

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; The eccentricity of the hyperbola is determined... more

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; The eccentricity of the ellipse is determined ... more

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; To determine the equation for the conic, follow... more

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; To determine the equation for the conic, follow... more

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; To determine the equation for the conic, follow... more

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; To determine the equation for the conic, follow... more

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; To solve follow these stepsThe value of is ... more

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; To solve follow these stepsThe value of is ... more

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; To solve follow these stepsThe value of is ... more

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; To solve follow these stepsThe value of is ... more

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; To solve follow these stepsThe value of is ... more

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  ; To solve follow these stepsThe value of is ... more

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; To solve follow these stepsThe figure shows ... more

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1 ; To solve follow these stepsThe figure shows ... more

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1.125 ; To solve follow these stepsThe figure ... more

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1.5 ; To solve follow these stepsThe figure shows ... more

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1 ; To solve follow these stepsThe figure shows ... more

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 {}  ; To solve follow these stepsUse elimination ... more

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{} ; To solve follow these stepsFirst multiply ... more

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{} ; To solve follow these stepsFirst multiply ... more

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{} ; To solve follow these stepsFirst multiply ... more

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{ ; To solve follow these stepsUse elimination ... more

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No solution ; To solve follow these stepsUse ... more

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{} ; To solve follow these stepsFirst multiply ... more

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The solutions are ordered pairs ; Adding both ... more

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No solution ; Multiplying 2 to the equation ; we ... more

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; Let be equation 1 and be equation 2.The ... more

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The solutions are ordered pairs ; Adding both ... more

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The solutions are ordered pairs ; Adding both ... more

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; From equation ; we get . ; Substituting in ... more

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; From equation ; . ; Substituting in equation... more

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. ; From equation ; . ; Substituting in equation... more

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; The equation corresponds to the general ... more

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txy-2-3-4-1-1-301-213-1250 ; Plot the curve by ... more

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txy-23-4-12-10121052-18 ; Plot the curve by ... more

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txy-263-13202113026-1 ; Plot the curve by joining ... more

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, in . ; Plot the curve by joining the points of... more

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, in . ; Plot the curve by joining the points of... more

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, in . ; Plot the curve by joining the points of... more

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, in . ; Plot the curve by joining the points of... more

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; As find the range of values of by ... more

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; As find the range of values of by ... more

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; As find the range of values of by ... more

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; As find the range of values of by ... more

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; As find the range of values of by ... more

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; As find the range of values of by ... more

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; As find the range of values of by ... more

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; As find the range of values of by ... more

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; As find the range of values of by ... more

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; Find the range of values of by substituting ... more

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; As , find the range of values of by ... more

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; Find the range of values of by substituting ... more

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; Find the range of values of by substituting ... more

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; Find the range of values of by substituting ... more

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; Find the range of values of by substituting ... more

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; As and the variable can attain any real value... more

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where and . ; Eliminate the parameter and find ... more

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where and . ; Eliminate the parameter and find ... more

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where and . ; Eliminate the parameter and find ... more

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; To convert the equation into two parametric ... more

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; To convert the equation into two parametric ... more

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; To convert the equation into two parametric ... more

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; To convert the equation into two parametric ... more

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; To convert the equation into two parametric ... more

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; To convert the equation into two parametric ... more

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; To convert the equation into two parametric ... more

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; To convert the equation into two parametric ... more

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; Time of flight of a projectile is given by ... more

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; Equation of a line that passes through a point... more

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; Equation of a hyperbola is given by Now to ... more

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Standard form and it represents an ellipse. ; ... more

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; hyperbola ; Use the completing square method to ... more

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; parabola ; Use the completing square method to ... more

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; First, write the given equation in standard ... more

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; First, write the given equation in standard ... more

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; First, write the given equation in standard ... more

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; The eccentricity of a parabola is always 1. So ... more

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; Since, foci are . Therefore, Distance between ... more

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; Since, foci are . Therefore, Distance between ... more

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; Multiply the equation (1) by 2. ; Add equations... more

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Ellipse:Compare equation with standard equation ... more

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; Write in term of . ; For finding an equivalent... more

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; domain: ; range: ; The standard equation of a ... more

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; domain: ; range: ; The standard equation of a ... more

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; domain: ; range: ; The standard equation of a ... more

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; 1 ; Use the completing square method to write ... more

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; 2 ; By using the method of completing square, ... more

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Center is at and radius is 1 unit. ; Transform ... more

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Center is at and radius is 2 units. ; Transform ... more

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Given equation is in general form of circle ... more

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If the parabola equation is in the form of it will... more

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Given parabola equation is in the form of and ... more

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Given parabola equation is in the form of and ... more

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; Given that parabola is opening right so it will... more

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; Given that vertex is at origin and focus is at... more

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; Given that vertex is at origin and focus is at... more

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; Given that parabola vertex is at originand is ... more

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; Given that vertex is at and focus is at Since ... more

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; Given that vertex is at and focus is at Since ... more

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Compare given ellipse equation with the standard ... more

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Compare given ellipse equation with the standard ... more

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Compare given hyperbola equation with the standard... more

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Compare given ellipse equation with the standard ... more

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Compare given ellipse equation with the standard ... more

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The domain of this function is and range is  . ... more

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The domain of this function is and range is    . ... more

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; The vertex is , therefore, this is an ellipse ... more

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; The intercept is , and focus . Therefore, this... more

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; The intercept is , and focus is . Therefore, ... more

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; The intercept is , therefore, this is an ... more

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; The focus is and the value of e is . Therefore... more

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; The focus is and eccentricity is , therefore, ... more

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Divide both sides by 36. This will give the ... more

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The circle has an eccentricity equal to 0.  ; ... more

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The center is and the vertices are . ; Convert ... more

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The standard form is and the center is and the ... more

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The equation in the standard form is and the ... more

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The equation in the standard form is and the ... more

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; The focus can be calculated from and .Insert ... more

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; This is an equation of hyperbola with ... more

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; The vertex is and intercepts axis twice ... more

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; First, consider the equation of asymptote and ... more

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; Distance between points is that is . First ... more

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; A hyperbola is defined as the set of all the  ... more

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  and ; Adding both the equation to calculate ... more

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and ; Using substitution method , substitute ... more

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Making a table for values by substituting values ... more

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-24-8-111000111248Making a table for values by ... more

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; Substituting the value of from the equation ... more

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  ; Substituting the value of from the equation ... more

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and and   ; Let for that is at initial stage, ... more

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Let for that is at initial stage, Substituting ... more

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Elliptical trajectory.  ; Substitute value of ... more

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approximately ; Substitute value of distance from ... more

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domain : Range : ; Assuming and substitute ... more

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; Arranging variable and aside and taking root ... more

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Hyperbola with the horizontal transverse axis.Foci... more

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Point graph at ; Complete the whole square or,... more

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center at  Major axis : horizontalFoci :Vertices... more

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Focus :Vertex: ; Firstly make a table using the ... more

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; Length of major axis, Thus, .Since, and So,Now... more

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Equation of ellipse = Height of arch 10 feet from ... more

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The graph of the first equation is a ellipse with ... more

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