The normal at a variable point P on an ellipse x2/a2 + y2/b2 = 1 of eccentricity e meets the - Maths - Conic Sections

Question

The normal at a variable point P on an ellipse
x2/a2 + y2/b2 = 1 of
eccentricity e meets the axes of the ellipse in Q and R, then locus
of the mid point of QR is a conic with eccentricity e' such that
:

1) e' is independent of e

2) e' = 1

3) e' = e

​4) e' = 1/e

Shivam Mishra · Accepted Answer

Dear Student,
Please find below the solution to the asked query:

We know that for x2a2+y2b2=1
i,  equation of normal at Pacosθ,bsinθ isaxsecθ-bycosecθ=a2-b2For x-axis y=0⇒axsecθ=a2-b2⇒x=a2-b2acosθHence Qa2-b2acosθ,0For y-axis, x=0⇒-bycosecθ=a2-b2⇒y=-a2-b2bsinθHence R0,-a2-b2bsinθLet mid point of QR be h,k whose locus is to be found⇒h,k=a2-b2acosθ+02,0+-a2-b2bsinθ2⇒h,k=a2-b22acosθ,-a2-b22bsinθ⇒h=a2-b22acosθ and k=-a2-b22bsinθ⇒cosθ=2aha2-b2 and sinθ=-2bka2-b2Square and add⇒4a2h2a2-b22+4b2k2a2-b22=1Hence locus is ellipse⇒4a2x2a2-b22+4b2y2a2-b22=1On comparing with x2A2+y2B2=1A2=a2-b224a2 and B2=a2-b224b2Note that if a2>b2 for i, then A2<B2 for obtained ellipsee'2=1-A2B2=1-a2-b224a2a2-b224b2=1-b2a2 which is e2 for i⇒e'2=e2⇒e'=e Answer

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The normal at a variable point P on an ellipse x2/a2 + y2/b2 = 1 of eccentricity e meets the axes of the ellipse in Q and R, then locus of the mid point of QR is a conic with eccentricity e' such that : 1). e' is independent of e 2). e' = 1 3). e' = e ​4). e' = 1/e

The normal at a variable point P on an ellipse x²/a² + y²/b² = 1 of eccentricity e meets the axes of the ellipse in Q and R, then locus of the mid point of QR is a conic with eccentricity e' such that :

1). e' is independent of e

2). e' = 1

3). e' = e

4). e' = 1/e