Question

Derive the equation of the parabola with a focus at (−5, −5) and a directrix of y = 7

Answer 1

Answer: The equation of the parabola is $(x+5)^2=-24(y-1)$.

Explanation:

It is given that the focus of the parabola is (-5,-5) and the directrix is y=7.

The standard form of the parabola is,

$(x-h)^2=4p(y-k)$

Where y=k-p is directrix and (h,k+p) is the focus.

Since focus is given,

$(h,k+p)=(-5,-5)$

On comparing,

$h=-5$

$k+p=-5$      .... (1)

The directrix is y=7.

$k+p=7$       .... (2)

Add equation (1) and (2),

$2k=2$

$k=1$

Put this value in (1).

$p=-6$

Put p= -6, h= -5 and k=1 in the standard form of the parabola.

$(x+5)^2=4(-6)(y-1)$

$(x+5)^2=-24(y-1)$

Therefore, the equation of parabola is $(x+5)^2=-24(y-1)$.

Answer 2

You can derive the formula which becomes...y=[1/(2(b-k))](x-a)^2+(1/2)(b+k)(a,b)=(-5,-5) and k=7 because y=7substitutey=[1/(2(-5-7))](x-[-5])^2+(1/2)(-5+7)y=[1/(-24)](x+5)^2+(1/2)(2)y=(-1/24)(x+5)^2+1 you can also do the following...the distance from the focus to a point is the same as the distance from the point to the directrix√(y-7)^2=√(x-[-5])^2+(y-[-5])^2square both sides and expandy^2-14y+49=x^2+10x+25+y^2+10y+25cancel each y^2 on both sides-14y+49=x^2+10x+25+10y+25-14y-10y=x^2+10x+25+25-49-24y=(x^2+10x+25)-24-24y=(x+5)^2-24divide both sides by -24y=(-1/24)(x+5)^2+1 again
so tell me if i helped