write in standard form, equations to represent each of the following conic sections

Question

Write in standard form, equations to represent each of the following conic sections

1. parabola, focus (-2,4) and directrix x=4
2. circle, endpoints of the diameter (3,2) and (-9,-8)
3. ellipse, vertices (endpoints of the major axis) (5,0) and (5,12) and endpoints of the minor axis (1,6) and (9,6)
4. hyperbola, vertices (4,2) and (-6,2) and foci (5,2) and (-7,2)

Answer 1

1.Vertex: ((-2 + 4)/2,4) = (1,4); 2p = -2 - 4 = -6; p = - 3

So, equation of parabola: (y - 4)² = -12(x-1)

2.circle, endpoints of the diameter (3,2) and (-9,-8)

Center ((3 - 9)/2,(2 - 8)/2) = (-3, - 3).

r² = (- 3 + 9)² + (-3 + 8)² = 36 + 25 = 61

Equation of circle: (x + 3)² + (y + 3)² = 61

3.ellipse, vertices (endpoints of the major axis) (5,0) and (5,12) and endpoints of the minor axis (1,6) and (9,6). Center is (5, 6); 2a = 12 - 0 = 12; a = 6; 2b = 9 - 1 = 8; b = 4.

Equation of ellipse: (x - 5)²/25 + (y - 6)²/16 = 1

4.hyperbola, vertices (4,2) and (-6,2) and foci (5,2) and (-7,2); 2a = 4 - (-6) = 10; a = 5; 2c = 5 - (-7) = 12; c = 6; b² = 6² - 5² = 11. Center: ((4 - 6)/2,2) = (-1, 2);

Equation of hyperbola:(x + 1)²/25 - (y - 2)²/11 = 1