CircleMedium
Question
The normal at a variable point P on an ellipse 
= 1 of eccentricity e meets the axes of the ellipse in Q and R then the locus of the mid-point of QR is a conic with an eccentricity e′ such that -
= 1 of eccentricity e meets the axes of the ellipse in Q and R then the locus of the mid-point of QR is a conic with an eccentricity e′ such that -Options
A.e′ is independent of e
B.e′ = 1
C.e′ = e
D.e′ = 1/e
Solution
Eqution of normal at P is
axsec θ - by cosec θ = a2 - b2 ...... (i)
Q ≡
,R ≡ 
Let middle point of QR be S(h, k).
2h =
. cosθ ; 2k = - 
. sinθ
2h = ae2cosθ k = -
sin θ
cos θ =
.... (ii) sinθ = 
..... (iii)
Square & add (ii) & (iii),
= 1
⇒
= 1 ⇒ 
= 1
where, A =
& B = 
B > A
e′ = 1 -
= 1 - 
= 1 -
= e2 ⇒ e′ = e
axsec θ - by cosec θ = a2 - b2 ...... (i)
Q ≡
,R ≡ Let middle point of QR be S(h, k).
2h =
. cosθ ; 2k = - . sinθ2h = ae2cosθ k = -
sin θcos θ =
.... (ii) sinθ = ..... (iii)Square & add (ii) & (iii),
= 1⇒
= 1 ⇒ = 1where, A =
& B = B > A
e′ = 1 -
= 1 - = 1 -
= e2 ⇒ e′ = eCreate a free account to view solution
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