Question

A parabola can be drawn given a focus of left bracket, minus, 6, comma, 9, right bracket(−6,9) and a directrix of y, equals, 3y=3. Write the equation of the parabola in any form.

Answer 1

Hi,

y=(x+6)²/12 +6

Explanation:

All point P(x,y) of the parabola is equal-distant to the focus F(a,b) and the directrix D (y=k)

`|PF|^2=(x-a)^2+(y-b)^2\\|PD|^2=(y-k)^2\\\\\\(x-a)^2+y^2-2by+b^2=y^2-2ky+k^2\\\\2y(b-k)=(x-a)^2+b^2-k^2\\\\\\\boxed{y=((x-a)^2)/(2(b-k)) +(b+k)/(2) }\\\\a=-6,b=9,k=3\\y=((x+6)^2)/(2*(9-3)) +(9+3)/(2) \\\\\\\boxed{y=((x+6)^2)/(12) +6} \\`