Question

Derive the equation of the parabola with a focus at (–5, 5) and a directrix of y = -1.

Answer 1

( x + 5 )^2 = - ( y - 5 )

Answer 2

`(x+5)^2=12(y-2)`

Explanation:

We are given that focus of parabola at (-5,5).

Equation of directrix y=-1

We have to derive the equation of parabola

We know that the parabola is the set of points (x,y) equally distant from (-5,5) and (x,-1).

Distance formula

`√((x_2-x_1)^2+(y_2-y_1)^2)`

Apply this formula

`√((x+5)^2+(y-5)^2)=√((x-x)^2+(y+1)^2)`

Squaring on both sides then we get

`(x+5)^2+(y-5)^2=(y+1)^2`

`(x+5)^2+y^2-10y+25=y^2+2y+1` (
`(a-b)^2=a^2+b^2-2ab,(a+b)^2=a^2+b^2+2ab`)

`(x+5)^2=y^2+2y+1-y^2+10y-25`

`(x+5)^2=12y-24`

`(x+5)^2=12(y-2)`

This is required equation of parabola along y- axis.