Question

1. which conic section is represented by the equation $36x^{2}+8y^{2}=81$?

circle
ellipse
parabola
hyperbola

Explanation:

Step1: Rewrite to standard conic form

Divide all terms by 81:
$$\frac{36x^2}{81} + \frac{8y^2}{81} = 1$$
Simplify fractions:
$$\frac{4x^2}{9} + \frac{8y^2}{81} = 1$$
Or
$$\frac{x^2}{\frac{9}{4}} + \frac{y^2}{\frac{81}{8}} = 1$$

Step2: Match to conic criteria

The equation fits the ellipse form $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$, where $a^2
eq b^2$ (eliminating circle), has two squared terms with positive coefficients (eliminating parabola and hyperbola).

Answer:

Ellipse