Question

What is the equation of the parabola that has a vertex at `(10, 8)` and passes through point `(5,83)`?

Answer 1

Actually, there are two equations that satisfy the specified conditions:

`y = 3(x - 10)^2+8` and `x = -1/1125(y-8)^2+10`

A graph of both parabolas and the points is included in the explanation.

Explanation:

There are two general vertex forms:

`y = a(x-h)^2+k` and `x = a(y-k)^2+h`

where `(h,k)` is the vertex

This gives us two equations where "a" is unknown:

`y = a(x - 10)^2+8` and `x = a(y-8)^2+10`

To find "a" for both, substitute the point `(5,83)`

`83 = a(5 - 10)^2+8` and `5 = a(83-8)^2+10`

`75 = a(-5)^2` and `-5 = a(75)^2`

`a=3` and `a = -1/1125`

The two equations are: `y = 3(x - 10)^2+8` and `x = -1/1125(y-8)^2+10`

Here is a graph that proves that both parabolas have the same vertex and intersect the required point: