QuestionIdentify the conic section for the equation $9 x^{2}+4 y^{2}=36$ : Ellipse, Parabola, Circle, or Hyperbola?

Studdy Solution

STEP 1

Assumptions1. The given equation is of the form $Ax^{} + By^{} = C$ . . We are asked to determine which conic section this equation represents.

STEP 2

The general form of the equation for different conic sections are as follows1. Circle $x^{2} + y^{2} = r^{2}$
2. Ellipse $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} =1$ , where $a \neq b$ . Hyperbola $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} =1$ or $\frac{y^{2}}{b^{2}} - \frac{x^{2}}{a^{2}} =1$
4. Parabola $y^{2} =4ax$ or $x^{2} =4ay$

STEP 3

We can rewrite the given equation in the standard form of a conic section by dividing the entire equation by36.
$\frac{9x^{2}}{36} + \frac{y^{2}}{36} =1$

STEP 4

implify the equation.
$\frac{x^{2}}{4} + \frac{y^{2}}{9} =1$

STEP 5

This equation is in the standard form of an ellipse, $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} =1$ , where $a^{2} =4$ and $b^{2} =9$ . Therefore, the given equation represents an ellipse.
The solution is Ellipse.

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