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) has the major axis is ( x − h ) 2 b 2 + ( y − k ) 2 a 2 = 1 , where 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 lengths 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 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 2 x 2 − 12 x + y 2 + 6 y + 26 = 0 . Compare the equation 2 x 2 − 12 x + y 2 + 6 y + 26 = 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 same sign, by the definition 2 stated above, the given conic may be an ellipse. Now, to complete the squares, rewrite 2 x 2 − 12 x + y 2 + 6 y + 26 = 0 as, 2 x 2 − 12 x + y 2 + 6 y + 26 = 0 2 ( x 2 − 6 x ) + ( y 2 + 6 y ) + 26 = 0 2 ( x 2 − 6 x + 9 − 9 ) + ( y 2 + 6 y + 9 − 9 ) + 26 = 0 2 ( x − 3 ) 2 + ( y + 3 ) 2 = − 26 + 27 Simplify further, 2 ( x − 3 ) 2 + ( y + 3 ) 2 = 1 ( x − 3 ) 2 1 2 + ( y + 3 ) 2 = 1 Thus, the equation of the conic is ( x − 3 ) 2 1 2 + ( y + 3 ) 2 = 1 (1) Thus, the equation (1) is of the form ( x − h ) 2 b 2 + ( y − k ) 2 a 2 = 1 . Therefore, the given conic 2 x 2 − 12 x + y 2 + 6 y + 26 = 0 is an ellipse

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ISBN: 8220102958371

Author: Watson

Publisher: CENGAGE L

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Chapter 11, Problem 51RE

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The graph of the equation 2x2−12x+y2+6y+26=0 is an ellipse, a parabola, or a hyperbola or a degenerate conic by complete the square.

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Chapter 11 Solutions

EBK PRECALCULUS: MATHEMATICS FOR CALCUL

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The graph of the equation 2 x 2 − 12 x + y 2 + 6 y + 26 = 0 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 2x2-12x+y2,+6y+26 = 0 is an ellipse, a parabola, or

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The graph of the equation 2x2−12x+y2+6y+26=0 is an ellipse, a parabola, or a hyperbola or a degenerate conic by complete the square.

Chapter 11 Solutions