# What are the equations in standard form of the equations 9x^2+16y^2=144 and 25x^2+9y^2-18y-216=0 ?

## (In ellipses. I don't know how to convert from general form to standard form.)

May 6, 2018

${x}^{2} / 16 + {y}^{2} / 9 = 1 \text{ }$ horizontal ellipse centered at $\left(0 , 0\right)$

${x}^{2} / 9 + {\left(y - 1\right)}^{2} / 25 = 1 \text{ }$ vertical ellipse centered at $\left(0 , 1\right)$

#### Explanation:

Given: $9 {x}^{2} + 16 {y}^{2} = 144$ and $25 {x}^{2} + 9 {y}^{2} - 18 y - 216 = 0$

To point an ellipse or hyperbola in standard form you must complete the square if needed, then divide by the constant on the right side:

Equation 1: $\text{ } \frac{9 {x}^{2}}{144} + \frac{16 {y}^{2}}{144} = \frac{144}{144}$

$\frac{9 {x}^{2}}{144} + \frac{16 {y}^{2}}{144} = 1$

Reduce so there isn't a constant in the numerator:

${x}^{2} / 16 + {y}^{2} / 9 = 1 \text{ }$ horizontal ellipse centered at $\left(0 , 0\right)$

It's horizontal because the largest denominator is on the $x$-term.

$- - - - - - - - - - - - - - - - - - - -$
Equation 2:
Requires completing of the square. First group $x$ terms together, $y$-terms together and put the constant on the right:

$\left(25 {x}^{2}\right) + \left(9 {y}^{2} - 18 y\right) = 216$

Factor: $25 {x}^{2} + 9 \left({y}^{2} - 2 y\right) = 216$

Complete the square on $y$ by halving the $- 2 y$ constant $= - 1$ and adding the 9(-1)^2 on the right that was added to the left when the square was completed:

$25 {x}^{2} + 9 {\left(y - 1\right)}^{2} = 216 + 9 {\left(- 1\right)}^{2}$

$25 {x}^{2} + 9 {\left(y - 1\right)}^{2} = 225$

Divide by the constant on the right side:

$\frac{25 {x}^{2}}{225} + \frac{9 {\left(y - 1\right)}^{2}}{225} = \frac{225}{225}$

$\frac{25 {x}^{2}}{225} + \frac{9 {\left(y - 1\right)}^{2}}{225} = 1$

Reduce so there isn't a constant in the numerator:

${x}^{2} / 9 + {\left(y - 1\right)}^{2} / 25 = 1 \text{ }$ vertical ellipse centered at $\left(0 , 1\right)$

It's vertical because the largest denominator is on the $y$-term.