Answer to Question #139819 in Analytic Geometry for Dhruv Rawat

Question #139819
Find the eccentricity, foci and centre of the conic 4x^2 + 9y^2= 36. Draw a rough sketch of it.
1
Expert's answer
2020-10-26T16:42:39-0400

4x2+9y2=36x29+y24=1(x0)29+(y0)24=1The equation describes an ellipsewith the origin as its centre.a2=9b2=4c2=a2b2=94=5c=±5The focus is(5,0),(5,0).Eccentricity=ca=53\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}\\


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