SITUATIONAL PROBLEMS INVOLVING CONIC SECTIONS

Question

Find all parabolas whose focus is a focus of the hyperbola 9x² -16y² +144x -320y -1600 = 0 and whose

directrix contains the top side of the hyperbola’s auxiliary rectangle.

Answer 1

9(x² + 16x + 64) - 576 - 16(y² + 20y + 100) + 1600 - 1600 = 0; 9(x + 8)² - 16(y + 10)² = 576;

(x + 8)²/16 - (y + 10)²/36 = 1;

Hyperbola: center at point (- 8, - 10); a = 4; b = 6, c² = 16 + 36 = 52; c = 2√13;

Foci: (- 2√13 - 8, - 10) and (2√13 - 8, - 10);

For parabola: y = 6 - 10 = - 4; y = - 4 is directrix; Now 2p = - 10 - (- 4) = - 6; p = - ;

Parabola with focus (- 2√13 - 8, - 10 vertex at point (- 2√13 - 8, - 7) and equation (x + 2√13 + 8)² = - 12(y + 7)

Parabola with focus (2√13 - 8, - 10 vertex at point (2√13 - 8, - 7) and equation (x - 2√13 + 8)² = - 12(y + 7)