(a) Explanation Definition used: Definition 1: “The equation of the shifted hyperbola with center as ( h , k ) and horizontal axis ( x -axis) axis as transverse axis is ( y − k ) 2 a 2 − ( x − h ) 2 b 2 = 1 , when a > 0 , b > 0 , c 2 = a 2 + b 2 ”. Definition 2: “The equation of the hyperbola with center at the origin, vertices ( 0 , − a ) , ( 0 , a ) and foci F 1 ( 0 , − c ) , F 2 ( 0 , c ) is y 2 a 2 − x 2 b 2 = 1 , where c = a 2 + b 2 , a > 0 , b > 0 ”. Definition 3: “The equation of the asymptotes of the hyperbola ( y − k ) 2 a 2 − ( x − h ) 2 b 2 = 1 are ( y − k ) a + ( x − h ) b = 0 and ( y − k ) a − ( x − h ) b = 0 ”. Definition 4: “If h and k are any two positive real numbers and a graph of any equation is in terms of x and y then. ( i ) Replacing x by x − h , shift the graph right to h units. ( i i ) Replacing x by x + h , shift the graph left to h units. ( i i i ) Replacing y by y − k , shift the graph upward to k units. ( i v ) Replacing y by y + k , shift the graph downward to k units”. Calculation: The given equation of conic is, 9 y 2 + 18 y = x 2 + 6 x + 18 . It can be written as follows. 9 y 2 + 18 y = x 2 + 6 x + 18 9 y 2 + 18 y − x 2 − 6 x = 18 9 ( y 2 + 2 y ) − ( x 2 + 6 x ) = 18 9 ( y 2 + 2 y + 1 − 1 ) − ( x 2 + 6 x + 9 − 9 ) = 18 Simplify further, 9 ( y 2 + 2 y + 1 ) − 9 − ( x 2 + 6 x + 9 ) + 9 = 18 9 ( y 2 + 2 y + 1 ) − ( x 2 + 6 x + 9 ) = 18 9 ( y + 1 ) 2 − ( x + 3 ) 2 = 18 9 ( y + 1 ) 2 18 − ( x + 3 ) 2 18 = 1 Thus, the equation is ( y + 1 ) 2 2 − ( x + 3 ) 2 18 = 1 (1) Now, by the definition 1, the equation (1) is a shifted hyperbola of y 2 2 − x 2 18 = 1 (2) That is, equation ( 1 ) may obtain from equation ( 2 ) replacing x by x + 3 and y by y + 1 . Therefore, first find the center, vertices and foci of the hyperbola ( 2 ) and then shift the x -coordinates by 3 units left and y -coordinates by 1 unit downward (b) To determine To sketch: The graph of the hyperbola 9 y 2 + 18 y = x 2 + 6 x + 18 .

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ISBN: 9781337037785

Author: Stewart

Publisher: CENGAGE C

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Chapter 11, Problem 35RE

(a)

To determine

To find: The center, vertices, foci and asymptotes of the hyperbola 9y2+18y=x2+6x+18 .

(b)

To determine

To sketch: The graph of the hyperbola 9y2+18y=x2+6x+18 .

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Chapter 11, Problem 34RE

Chapter 11, Problem 36RE

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Ch. 11.1 - A parabola is the set of all points in the plane...Ch. 11.1 - The graph of the equation x2 = 4py is a parabola...Ch. 11.1 - The graph of the equation y2 = 4px is a parabola...Ch. 11.1 - Label the focus, directrix, and vertex on the...Ch. 11.1 - Graphs of Parabolas Match the equation with the...Ch. 11.1 - Prob. 6E Ch. 11.1 - Graphs of Parabolas Match the equation with the...Ch. 11.1 - Prob. 8E Ch. 11.1 - Graphs of Parabolas Match the equation with the...Ch. 11.1 - Graphs of Parabolas Match the equation with the...

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The center, vertices, foci and asymptotes of the hyperbola 9 y 2 + 18 y = x 2 + 6 x + 18 .

Solution Summary: The author explains the center, vertices, foci, and asymptotes of the hyperbola.

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To find: The center, vertices, foci and asymptotes of the hyperbola 9y2+18y=x2+6x+18 .

To sketch: The graph of the hyperbola 9y2+18y=x2+6x+18 .

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Chapter 11 Solutions