Question

-4. The point of intersection of the tangents at the point \( P \) on the ellipse \( \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 \) and corresponding point \( Q \) on the auxiliary circle, lies on the line:

Answer 1

Let the point P be (acosθ , bsinθ)

Equation of tangent to x²​/ a² + y²​/b² =1 at P is T=0

bxcosθ + aysinθ=ab     ......(i)

Equation of auxiliary circle of ellipse is x²+y²=a²

Point corresponding to P on auxiliary circle is Q(acosθ,asinθ)

Equation of tangent to the circle at Q is T=0

xcosθ + ysinθ = a

⇒xcosθ = a−ysinθ 

⇒x = cosθa−ysinθ​

Substituting x in (i)

⇒bcosθ(cosθa−ysinθ​) + aysinθ = ab 

⇒ab−bysinθ + aysinθ = ab

Thus (a−b)ysinθ = 0

⇒y=0