Question

What is the standard form of the equation of the parabola with a directrix at x=-5 and a focus at (-7,-5)?

Answer 1

The equation of the parabola is `(y+5)^2=-4x-24=-4(x+6)`

Explanation:

Any point `(x,y)` on the parabola is equidistant from the directrix and the focus.

Therefore,

`x-(-5)=sqrt((x-(-7))^2+(y-(-5))^2)`

`x+5=sqrt((x+7)^2+(y+5)^2)`

Squaring and developing the `(x+7)^2` term and the LHS

`(x+5)^2=(x+7)^2+(y+5)^2`

`x^2+10x+25=x^2+14x+49+(y+5)^2`

`(y+5)^2=-4x-24=-4(x+6)`

The equation of the parabola is `(y+5)^2=-4x-24=-4(x+6)`

graph{((y+5)^2+4x+24)((x+7)^2+(y+5)^2-0.03)(y-100(x+5))=0 [-17.68, 4.83, -9.325, 1.925]}