Explanation Consider the second-degree equation A x 2 + B x + C y 2 + D y = E Case 1: Parabola The equation A x 2 + B x + C y 2 + D y = E represents a parabola if either A = 0 or C = 0 since then, the equation will either contain an x 2 term or y 2 term which suggests that the equation represents a parabola. Case 2: Circle The equation A x 2 + B x + C y 2 + D y = E represents a circle if A = C as since then, the coefficients of terms x 2 and y 2 will be same which suggests that the equation represents a circle. Case 3: Ellipse The equation A x 2 + B x + C y 2 + D y = E represents an ellipse if both A and C have the same sign, either positive or negative, and A ≠ C as since then, the coefficients of terms x 2 and y 2 will be different, but of the same sign which suggests that the equation represents an ellipse. Case 4: Hyperbola The equation A x 2 + B x + C y 2 + D y = E represents a hyperbola if both A and C have different signs, that is, exactly one of them is positive or negative, as since then, the coefficients of terms x 2 and y 2 will be of different signs which suggests that the equation represents a hyperbola. Now, consider the equation y 2 − x 2 + 2 x = 2 Compare the above equation with the standard equation A x 2 + B x + C y 2 + D y = E and see that A = − 1 , B = 2 , C = 1 , D = 0 and E = 2 . Here, A and C have different signs and A ≠ C and hence, the equation represents a hyperbola. Consider the equation 4 y 2 + x 2 − 2 x = 15 Compare the above equation with the standard equation A x 2 + B x + C y 2 + D y = E and see that A = 1 , B = − 2 , C = 4 , D = 0 and E = 15 . Here, A and C have the same sign and A ≠ C and hence, the equation represents an ellipse. Consider the equation y − x 2 = 2 x Rewrite this equation as: y − x 2 − 2 x = 0 Compare the above equation with the standard equation A x 2 + B x + C y 2 + D y = E and see that A = − 1 , B = − 2 , C = 0 , D = 1 and E = 0 . Here, C = 0 , that is, the equation has only an x 2 term and hence, the equation represents a parabola. Consider the equation y 2 + x 2 − 2 x = 0 Compare the above equation with the standard equation A x 2 + B x + C y 2 + D y = E and see that A = 1 , B = − 2 , C = 1 , D = 0 and E = 0 . Here, A and C have the same sign and A = C and hence, the equation represents a circle. Consider the equation 25 x 2 = 2500 − 25 y 2 Rewrite the above equation as, 25 x 2 + 25 y 2 = 2500 x 2 + y 2 = 100 Compare the above equation with the standard equation A x 2 + B x + C y 2 + D y = E and see that A = 1 , B = 0 , C = 1 , D = 0 and E = 100 . Here, A and C have the same sign and A = C and hence, the equation represents a circle. Consider the equation 100 x 2 = 25 y 2 + 2500 Rewrite the above equation as, 100 x 2 − 25 y 2 = 2500 4 x 2 − y 2 = 100 Compare the above equation with the standard equation A x 2 + B x + C y 2 + D y = E and see that A = 4 , B = 0 , C = − 1 , D = 0 and E = 100 . Here, A and C have different signs and A ≠ C and hence, the equation represents a hyperbola

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ISBN: 9780321916600

Author: Mark Dugopolski

Publisher: PEARSON

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Chapter 7.3, Problem 62E

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The conditions on A, B, C, D and E that will determine whether the equation Ax2+Bx+Cy2+Dy=E represents a parabola, an ellipse, a circle or a hyperbola.

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Chapter 7.3, Problem 61E

Chapter 7.3, Problem 63E

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College Algebra (6th Edition)

Ch. 7.1 - True or False? Explain. A parabola with focus (0,...Ch. 7.1 - Prob. 2FT Ch. 7.1 - Prob. 3FT Ch. 7.1 - Prob. 4FT Ch. 7.1 - Prob. 5FT Ch. 7.1 - Prob. 6FT Ch. 7.1 - Prob. 7FT Ch. 7.1 - Prob. 8FT Ch. 7.1 - Prob. 9FT Ch. 7.1 - Prob. 10FT

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The conditions on A, B, C, D and E that will determine whether the equation A x 2 + B x + C y 2 + D y = E represents a parabola, an ellipse, a circle or a hyperbola.

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The conditions on A, B, C, D and E that will determine whether the equation Ax2+Bx+Cy2+Dy=E represents a parabola, an ellipse, a circle or a hyperbola.

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