Answer to Question #139819 in Analytic Geometry for Dhruv Rawat

Question

Find the eccentricity, foci and centre of the conic 4x^2 + 9y^2= 36. Draw a rough sketch of it.

Answer 1

$\displaystyle 4x^2 + 9y^2 = 36\\ \frac{x^2}{9} + \frac{y^2}{4} = 1\\ \frac{(x - 0)^2}{9} + \frac{(y - 0)^2}{4} = 1\\ \textsf{The equation describes an ellipse}\\ \textsf{with the origin as its centre.}\\ a^2 = 9 \\ b^2 = 4\\ c^2 = a^2 - b^2 = 9 - 4 = 5\\ c = \pm\sqrt{5}\\ \therefore \textsf{The focus is}\, (\sqrt{5}, 0), (-\sqrt{5}, 0).\\ \textsf{Eccentricity}\, = \frac{c}{a} = \frac{\sqrt{5}}{3}\\$