Question

What is the equation in standard form of the parabola with a focus at (-10,-9) and a directrix of y= -4?

Answer 1

The equation of parabola is `y = -1/10(x+10)^2 -6.5`

Explanation:

The focus is at `(-10 , -9)` Directrix: `y =-4`. Vertex is at mid point between focus and directrix. So vertex is at `(-10, (-9-4)/2) or (-10, -6.5)` and the parabola opens downwards (a = -ive )

The equation of parabola is `y= a(x-h)^2 =k or y = a(x -(-10))^2+ (-6.5) or y= a(x+10)^2 -6.5` where `(h,k)` is vertex.

The distance between vertex and directrix , `d=6.5-4.0=2.5 = 1/(4|a|) :. a = -1/(4*2.5) = -1/10`

Hence the equation of parabola is `y = -1/10(x+10)^2 -6.5` graph{ -1/10(x+10)^2 - 6.5 [-40, 40, -20, 20]} [Ans]