Question

Example 10 Find the coordinates of the foci, the vertices, the lengths of major and minor axes and the eccentricity of the ellipse 9x? + 4v2 = 36.

JEE/Engineering Exams

Maths

Solution

100

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[ frac{x^{2}}{4}+frac{y^{2}}{9}=1 ] since the denominator of ( frac{y^{2}}{9} ) is larger than the denominator of ( frac{x^{2}}{4} ). the major axis is along the ( y ) -axis. Comparing the given equation with the standard equation ( frac{x^{2}}{b^{2}}+frac{y^{2}}{a^{2}}=1, ) we have ( b=2 ) and ( a=3 ) ( mathrm{Also} ) [ c=sqrt{a^{2}-b^{2}}=sqrt{9-4}=sqrt{5} ] and [ e=frac{c}{a}=frac{sqrt{5}}{3} ] Hence the foci are ( (0, sqrt{5}) ) and ( (0,-sqrt{5}), ) vertices are (0,3) and ( (0,-3), ) length of the major axis is 6 units, the length of the minor axis is 4 units and the eccentricity of the ellipse is ( frac{sqrt{5}}{3} )