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

May 28, 2017

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

$y = 3 {\left(x - 10\right)}^{2} + 8$ and $x = - \frac{1}{1125} {\left(y - 8\right)}^{2} + 10$

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

#### Explanation:

There are two general vertex forms:

$y = a {\left(x - h\right)}^{2} + k$ and $x = a {\left(y - k\right)}^{2} + h$

where $\left(h , k\right)$ is the vertex

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

$y = a {\left(x - 10\right)}^{2} + 8$ and $x = a {\left(y - 8\right)}^{2} + 10$

To find "a" for both, substitute the point $\left(5 , 83\right)$

$83 = a {\left(5 - 10\right)}^{2} + 8$ and $5 = a {\left(83 - 8\right)}^{2} + 10$

$75 = a {\left(- 5\right)}^{2}$ and $- 5 = a {\left(75\right)}^{2}$

$a = 3$ and $a = - \frac{1}{1125}$

The two equations are: $y = 3 {\left(x - 10\right)}^{2} + 8$ and $x = - \frac{1}{1125} {\left(y - 8\right)}^{2} + 10$

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