Question

Find the standard form:
Focus: (4, -9)
Directrix: x=2

Answer 1

I'm guessing the question is asking the standard form of a "parabola" when given this information.
You're given that the focus has coordinates of:
\[\large S(4, -9)\]
and the equation of the Directrix:
\[\large D:x=2\]
Okay, the vertex of the parabola is in between the directrix and the focus.
So the coordinates of the vertex would be (3, -9)
Okay. The general formula for this "side-ways parabola" would be:
\[\large (y-k)^2=4a(x-h)\]
where:(h, k) is the vertex.
And a is the length between the vertex and the directrix/focus.
It could of been -4a(x-h) but as you can see, the focus is to the right of the directrix which means that it must be a "positive side-ways parabola".
The value of a would be 1 since the vertex is 1 unit apart from the directrix and the focus.
So now we have all the values we need.
The standard form of that parabola would be:
\[\large (x-(3))^2=4(1)[y-(-9)]\]
\[\large (x-3)^2=4(y+9)\]