The correct option is B The locus of mid-point of QR is a conic whose eccentricity is same as of given ellipse
x2a2+y2b2=1, (a>b)
Equation of normal at point P(acosθ,bsinθ) is,
axcosθ−bysinθ=a2−b2
It meets axes at Q((a2−b2)cosθa,0) and R(0,−(a2−b2)sinθb)
Let M(h,k) be the midpoint of QR.
Then 2h=(a2−b2)cosθa and 2k=−(a2−b2)sinθb
⇒cos2θ+sin2θ=4h2a2(a2−b2)2+4k2b2(a2−b2)2=1
Locus is x2(a2−b22a)2+y2(a2−b22b)2=1 which represents an ellipse.
Since a2e2=a2−b2,
∴x2(a4e44a2)+y2(a4e44b2)=1
Since a>b,
∴a4e44b2>a4e44a2e′=
⎷1−(a4e44a2)(a4e44b2)=√1−b2a2=e⇒e′=e