Question

What is the standard form of the equation of the parabola with a directrix at x=23 and a focus at (5,5)?

Answer 1

The equation of parabola will be: `(y-5)^2=-36(x-14)`

Explanation:

Given equation of directrix of parabola is `x=23` & the focus at `(5, 5)`. It is clear that it is a horizontal parabola with sides diverging in -ve x-direction. Let general equation of parabola be
`(y-y_1)^2=-4a(x-x_1)` having equation of directrix: `x=x_1+a` & the focus at `(x_1-a, y_1)`
Now, comparing with given data, we have `x_1+a=23`, `x_1-a=5, y_1=5` which gives us `x_1=14, a=9` hence the equation of parabola will
`(y-5)^2=-4\cdot 9(x-14)`
`(y-5)^2=-36(x-14)`