wiz-icon
MyQuestionIcon
MyQuestionIcon
1
You visited us 1 times! Enjoying our articles? Unlock Full Access!
Question

The normal at a variable point P on ellipse x2a2+y2b2=1,(a>b) meets axes of the ellipse at Q and R. Then which of the following is/are CORRECT?

A
The locus of mid-point of QR is an ellipse
Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses
B
The locus of mid-point of QR is a conic whose eccentricity is same as of given ellipse
Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses
C
The locus of mid-point of QR is a hyperbola
No worries! We‘ve got your back. Try BYJU‘S free classes today!
D
The locus of mid-point of QR is a conic with its eccentricity half of the eccentricity of given ellipse
No worries! We‘ve got your back. Try BYJU‘S free classes today!
Open in App
Solution

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,
axcosθbysinθ=a2b2
It meets axes at Q((a2b2)cosθa,0) and R(0,(a2b2)sinθb)

Let M(h,k) be the midpoint of QR.
Then 2h=(a2b2)cosθa and 2k=(a2b2)sinθb
cos2θ+sin2θ=4h2a2(a2b2)2+4k2b2(a2b2)2=1
Locus is x2(a2b22a)2+y2(a2b22b)2=1 which represents an ellipse.

Since a2e2=a2b2,
x2(a4e44a2)+y2(a4e44b2)=1

Since a>b,
a4e44b2>a4e44a2e=     1(a4e44a2)(a4e44b2)=1b2a2=ee=e

flag
Suggest Corrections
thumbs-up
1
Join BYJU'S Learning Program
similar_icon
Related Videos
thumbnail
lock
Line and Ellipse
MATHEMATICS
Watch in App
Join BYJU'S Learning Program
CrossIcon