The point of intersection of the tangents at the point P on the ellipsex2a2+y2b2=1 and its corresponding point Q on the auxiliary circle, lies on the line
y = 0
Let the point P be (a cosθ, b sin θ) with eccentric angle θ. The corresponding point Q on the auxiliary circle will be (a cosθ, b sin θ).We know that
the equation of auxiliary circle of x2a2+y2b2=1 is x2 + y2 = a2 we will now find the tangents and find their intersection.
Tangent to ellipse:
(a cosθ, b sin θ) is the point ...(1)
tangent to circle:
(a cosθ, a sin θ) is the point
x2 + y2 = a2 or x2a2+y2b2=1 is the circle. ...(2)
(1)-(2) ⇒
⇒ y = 0
So all the tangents intersect on x - axis or y = 0.
⇒ Option (C).