If the tangents at any point P on the ellipse x2a2+y2b2=1 meets the tangents at the vertices A and A′ in L and L′ respectively, then AL.A′L′=
Let tangent is drawn at any point P(acosθ,bsinθ) on the ellipse x2a2+y2b2=1 then equation of tangent is
xacosθ+ybsinθ=1......(i)
Equation of tangent at A(a,0) is x=a
Put x=a in (i)
⇒aacosθ+ybsinθ=1⇒y=(1−cosθ)bsinθ⇒AL=(1−cosθ)bsinθ
Equation of tangent at A′(−a,0) is x=−a
Put x=−a in (i)
⇒−aacosθ+ybsinθ=1⇒y=(1+cosθ)bsinθ⇒A′L′=(1+cosθ)bsinθ
AL.A′L′=((1−cosθ)bsinθ)((1+cosθ)bsinθ)AL.A′L′=(1−cos2θ)b2sin2θAL.A′L′=b2
Hence, option D is correct.