# How do you write a standard form equation for the hyperbola with 36x^2 - 100y^2 - 72x + 400y = 3964?

Mar 6, 2016

$\frac{{\left(x - \frac{3}{2}\right)}^{2}}{\frac{47 \sqrt{3}}{6}} ^ 2 - \frac{{\left(y - 2\right)}^{2}}{\frac{47 \sqrt{2}}{10}} ^ 2 = 1$

#### Explanation:

The standard form looks something like this

$\frac{{\left(x - h\right)}^{2}}{{a}^{2}} - \frac{{\left(y - k\right)}^{2}}{{b}^{2}} = 1$,

where $h$, $k$, $a$ and $b$ are constants to be determined.

First thing to do is to simplify the equation. Divide everything by 4.

$6 {x}^{2} - 25 {y}^{2} - 18 x + 100 y = 991$

Next, factorize.

$6 \left({x}^{2} - 3 x\right) - 25 \left({y}^{2} - 4 y\right) = 991$

Next, complete the square

$6 {\left(x - \frac{3}{2}\right)}^{2} - 25 {\left(y - 2\right)}^{2} = 991 + 6 {\left(\frac{3}{2}\right)}^{2} + 25 {\left(2\right)}^{2}$

$= \frac{2209}{2}$

Divide both sides by 2209/2 to get 1 on the right hand side.

$\frac{{\left(x - \frac{3}{2}\right)}^{2}}{\frac{2209}{12}} - \frac{{\left(y - 2\right)}^{2}}{\frac{2209}{50}} = 1$

Take the square root of the denominator, and square it afterwards to make it into standard form.

$\frac{{\left(x - \frac{3}{2}\right)}^{2}}{\frac{47 \sqrt{3}}{6}} ^ 2 - \frac{{\left(y - 2\right)}^{2}}{\frac{47 \sqrt{2}}{10}} ^ 2 = 1$