Question

# The tangent at any point on the ellipse x2a2+y2b2=1 meets the auxiliary circle at two points which subtend a right angle at the centre. When the eccentricity of the ellipse is minimum then

A
the y intercept made by the tangent is y=±b2 when a>b
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B
the x intercept made by the tangent is x=±a2 when a<b
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C
minimum value of eccentricity is 12
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D
When a>b, then the area enclosed between the tangents and the ellipse is (4π)ab
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Solution

## The correct options are C minimum value of eccentricity is 1√2 D When a>b, then the area enclosed between the tangents and the ellipse is (4−π)ab When a>b Equation of the Auxiliary circle is, x2+y2=a2 ⋯(1) Equation of tangent at any point P(acosα,bsinα) is, xcosαa+ysinαb=1 ⋯(2) Making equation (2) homogeneous using equation (1), x2+y2−a2(xcosαa+ysinαb)2=0(1−cos2α)x2+(1−a2b2sin2α)y2 −(2absinαcosα)xy=0 ⋯(3) Equation (3) represents the combine equation of the line OL and OM, Since, ∠LOM=π2 Coefficient of x2+Coefficient of y2=0 ⇒(1−cos2α)+(1−a2b2sin2α)=0⇒sin2α(a2b2−1)=1⇒sin2α(11−e2−1)=1[∵b2=a2(1−e2)]⇒e=1√1+sin2α When b>a Equation of the Auxiliary circle is, x2+y2=b2 ⋯(1) Equation of tangent any point P(acosα,bsinα) xcosαa+ysinαb=1 ⋯(2) Making equation (2) homogeneous using equation (1), x2+y2−b2(xcosαa+ysinαb)2=0 Coefficient of x2+Coefficient of y2=0 ⇒1−b2a2cos2α+cos2α=0⇒cos2α(b2a2−1)=1⇒cos2α(11−e2−1)=1[∵a2=b2(1−e2)]⇒e=1√1+cos2α Therefore, a>b⇒e=1√1+sin2αb>a⇒e=1√1+cos2α Minimum value of eccentricity is e=1√2 When a>b α=π2,3π2 Tangents y=±b When b>a α=0,π Tangents x=±a Area bounded when a>b, A=2a×2b−πab=(4−π)ab

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