# How do you identify the following equation 25x^2 + 4y^2 = 100 as a circle, parabola, ellipse or hyperbola?

Dec 9, 2015

Explanation below
- Ellipse, $\frac{{x}^{2}}{4} + \frac{{y}^{2}}{25} = 1$

#### Explanation:

The shape is

Case 1) A parabola, if only ONE variable is square

Case 2) A circle, if the coefficients of the variables are $1$, both variables are squares, and addition of the variables gives any number other than $1$.

Case 3) An ellipse, if the coefficients are anything other than $1$, both variables are squares, and addition of the variables gives $1$.

Case 4) A hyperbola, if both variables are squares, and subtraction of the variables gives $1$.

For this equation, $25 {x}^{2} + 4 {y}^{2} = 100$,

We have:

1. Both variables are squares.
2. The coefficients are other than $1$.

We can divide both sides by $100$, to get the equation equal to $1$:

$\frac{25 {x}^{2}}{100} + \frac{4 {y}^{2}}{100} = \frac{100}{100}$

$\implies \frac{{x}^{2}}{4} + \frac{{y}^{2}}{25} = 1$

This satisfies Case 3 - It's an ellipse.