An ellipse centered at 2 3 with a major vertical axis 14 uni

An ellipse centered at (-2, 3) with a major vertical axis 14 units long and a minor axis 6 units long. A parabola with vertex at (5, 0) and directrix at x = 7.

Solution

ellipse centre = ( -2 , 3)

major axis = 14

minor axis = 6

standard equation of ellipse is

(x-h)^2 / a^2 + (y-k)^2 / b^2 = 1

where, h,k = centre

a = semi major axis

b = semi minor axis

2a = 14

a = 7

2b = 6

b = 3

plugging the values in standrad equation

( x + 2)^2 / 7^2 + ( y - 3)^2 / 3^2 = 1

(x+2)^2/ 49 + (y-3)^2 / 9 = 1

8) vertex = 5 , 0

directrix x = 7

standard equation of parabola is

(y-k)^2 = 4p ( x-h )

where , h,k = vertex

parabola opens left

equation of parabola is

(y-0)^2 = 4p (x-5)

p = -2 (distance between directrix and vertex)

4p = -8

therefore equation is

y^2 = -8 (x-5)