Problems Involving Conic Sections

Question

Find the standard equation of the hyperbola whose conjugate axis is on the directrix of the parabola y² -5x+12y=-16 , having the focus of the parabola as one of its foci, and the vertex of the parabola as one of its vertices. 

Answer 1

(y + 6)² = 5(x + 4). Vertex of parabola (- 4, - 6). 4p = 5; p = 5/4. Focus of parabola at point (- 11/4, - 6) and equation of directrix x = - 21/4. So center of hyperbola at point (- 21/4, - 6). Now , c = - 11/4 + 21/4 = 5/2 and a = 5/4. Then b² = 25/4 - 25/16 = 75/16. We have equation of hyperbola:

(x + 21/4)²/(25/16) - (y + 6)²'(75/16) = 1.

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