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 shifted ellipse with center ( h , k ) and vertical axis ( y -axis) axis has the major axis is ( x − h ) 2 b 2 + ( y − k ) 2 a 2 = 1 , when a > b > 0 , c 2 = a 2 − b 2 ”. Definition 4: “The equation of the ellipse with center at the origin, vertices V ( 0 , ± a ) and foci F ( 0 , ± c ) is x 2 b 2 + y 2 a 2 = 1 , where c = a 2 − b 2 , a > b > 0 .” Definition 5: “The length of major and minor axes of the ellipse x 2 b 2 + y 2 a 2 = 1 or ( x 2 a 2 + y 2 b 2 = 1 ) are 2 a and 2 b respectively.” Definition 6: “If h and k are any two positive real numbers and a graph of any equation is interms 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 conic equation is 2 x 2 + y 2 = 2 y + 1 . Compare the equation 2 x 2 + y 2 = 2 y + 1 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 same sign, by the definition 2 stated above, the given conic may be an ellipse. Now, to complete the squares, rewrite 2 x 2 + y 2 − 2 y − 1 = 0 as, 2 x 2 + y 2 − 2 y + 1 − 1 = 1 2 x 2 + ( y − 1 ) 2 = 2 2 x 2 2 + ( y − 1 ) 2 2 = 1 x 2 + ( y − 1 ) 2 2 = 1 Thus, equation x 2 + ( y − 1 ) 2 2 = 1 is of the form ( x − h ) 2 b 2 + ( y − k ) 2 a 2 = 1 . That is, given conic 2 x 2 + y 2 = 2 y + 1 is an ellipse. By the definition 3 stated above, equation (1) is a shifted ellipse of x 2 + y 2 2 = 1 . Compare the equation x 2 + y 2 2 = 1 with the ellipse x 2 b 2 + y 2 a 2 = 1 , a 2 = 2 a = 2 b 2 = 1 b = 1 Clearly a = 2 , b = 1 and a > b

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Chapter 11.4, Problem 52E

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The graph of the equation 2x2+y2=2y+1 is an ellipse, a parabola, or a hyperbola or a degenerate conic by complete the square.

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Chapter 11.4, Problem 51E

Chapter 11.4, Problem 53E

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Chapter 11 Solutions

PRECALCULUS LL + WEBASSIGN/COREQUISITE

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The graph of the equation 2 x 2 + y 2 = 2 y + 1 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+y2,=2y+1 is an ellipse.

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The graph of the equation 2x2+y2=2y+1 is an ellipse, a parabola, or a hyperbola or a degenerate conic by complete the square.

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