Explanation Definition used: Definition 1: “The graph of the general equation of shifted conic A x 2 + C y 2 + D x + E y + F = 0 , where A and C are not both 0 is conic or a degenerate conic”. Definition 2: “The graph of the general equation of shifted conic A x 2 + C y 2 + D x + E y + F = 0 , where A and C are not both 0 is a non-degenerate conic. Then it represents (i) A parabola if A or C is 0 . (ii) An ellipse if A and C have the same sign. (iii) A circle if A = C . (iv) A hyperbola if A and C have opposite sign”. Definition 3: “The equation of the hyperbola with center at the origin, vertices V ( 0 , ± a ) and foci F ( 0 , ± c ) is y 2 a 2 − x 2 b 2 = 1 , where c = a 2 + b 2 , a > b > 0 ”. Definition 4: “The equation of the asymptotes of the hyperbola y 2 a 2 − b 2 b 2 = 1 are y a + x b = 0 and y a − x b = 0 ”. Definition 5: “The equation of the asymptotes of the hyperbola ( x − h ) 2 a 2 − ( y − k ) 2 b 2 = 1 are ( x − h ) a + ( y − k ) b = 0 and ( x − h ) a − ( y − k ) b = 0 ”. Definition 6: “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: Given equation of conic is 2 x 2 + 4 = 4 x + y 2 . It can be written as 2 x 2 − y 2 − 4 x + 4 = 0 . Compare the equation 2 x 2 − y 2 − 4 x + 4 = 0 with A x 2 + C y 2 + D x + E y + F = 0 . A = 2 , C = − 1 . Since both A and C are not equate to zero and of the opposite signs, by the definition 2 stated above, the given conic may be a hyperbola. Now, to complete the squares, rewrite 2 x 2 + 4 = 4 x + y 2 as, 2 x 2 − 4 x − y 2 + 4 = 0 2 ( x 2 − 2 x ) − y 2 + 4 = 0 2 ( x 2 − 2 x + 1 − 1 ) − y 2 = − 4 2 ( x − 1 ) 2 − y 2 = − 4 + 2 Simplify further, 2 ( x − 1 ) 2 − y 2 = − 2 2 ( x − 1 ) 2 − 2 − y 2 − 2 = − 2 − 2 y 2 2 − ( x − 1 ) 2 = 1 Thus, the equation of the conic is y 2 2 − ( x − 1 ) 2 = 1 (1) Thus, equation (1) is of the form ( y − k ) 2 a 2 − ( x − h ) 2 b 2 = 1

Precalculus: Mathematics for Calculus (Standalone Book)

7th Edition

ISBN: 9781305071759

Author: James Stewart, Lothar Redlin, Saleem Watson

Publisher: Brooks Cole

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Chapter 11, Problem 50RE

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The graph of the equation 2x2+4=4x+y2 is an ellipse, a parabola, or a hyperbola or a degenerate conic by complete the square.

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Chapter 11, Problem 49RE

Chapter 11, Problem 51RE

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Chapter 11 Solutions

Precalculus: Mathematics for Calculus (Standalone Book)

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 graph of the equation 2 x 2 + 4 = 4 x + y 2 is an ellipse, a parabola, or a hyperbola or a degenerate conic by complete the square.

Solution Summary: The author explains that the graph of the equation 2x2+4=4x+y is a hyperbola.

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The graph of the equation 2x2+4=4x+y2 is an ellipse, a parabola, or a hyperbola or a degenerate conic by complete the square.

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Chapter 11 Solutions