Question

What is the standard form of the equation of the parabola with a directrix at x=6 and a focus at (9,5)?

Answer 1

`6x=y^2-10y+70`

Explanation:

Parabola is the locus of a point which moves so that its distance from a given point called focus and a given line called directrix is always same.

Let the point on parabola be `(x,y)`. Here focus is `(9,5)` and its distance from focus is `sqrt((x-9)^2+(y-5)^2)`.

And as directrix is `x=6` and distance of `(x,y)` from `x=6` is `|x-6|`. Hence equation of parabola is

`(x-9)^2+(y-5)^2=(x-6)^2`

or `x^2-18x+81+y^2-10y+25=x^2-12x+36`

or `cancelx^2+81+y^2-10y+25-36=cancelx^2+18x-12x`

or `6x=y^2-10y+70`

or `x=1/6(y^2-10y+70)`

graph{(y^2-10y+70-6x)(x-6)((x-9)^2+(y-5)^2-0.03)=0 [-0.83, 19.17, -0.36, 9.64]}