# Parabola

#### Yohib

##### New member
Prove that the focal radius of point on (a,b) of equation of parabola x^2=4ay is |a + b|

#### Subhotosh Khan

##### Super Moderator
Staff member
Prove that the focal radius of point on (a,b) of equation of parabola x^2=4ay is |a + b|
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#### HallsofIvy

##### Elite Member
I suggest that the first thing you need to do is state exactly what "focal radius of a point" means.

#### HallsofIvy

##### Elite Member
Yes (though I would have said "focal distance"), that is the distance from the vertex of the parabola to its focus. However, the original post specifically asked for the focal radius "of point (a, b)" and I still don't know what that means.

#### Dr.Peterson

##### Elite Member
My concern is that the question seems to be using "a" in two different ways: an x-coordinate, and a parameter in the equation.

#### Cubist

##### Full Member
I've actually completed this question - and the proof does work out. I'm unsure whether or not to post my work. I don't want to just give away the whole result to OP. But on the other hand they don't seem to be responding and so they may have solved it.

@Dr.Peterson I treated "a" as a constant within the equation. And for a certain value of "a" you obtain a single point (a,b) where "b" can be expressed in terms of a (this helps with the proof).

The second link in post#4 almost shows the identical equation to the question, and it states where the focus is. Given this, the rest is actually easy.

(Before I found this link I'd worked out the location of the focus "the hard way" by using angle of incidence=angle of reflection. I guess this is what most people would do in an exam since I'm guessing that not many people would have the parabola focus position memorised.)