1
You visited us 1 times! Enjoying our articles? Unlock Full Access!
Question

# If a straight line through the point P(λ,2) where λ≠0, meets the ellipse x29+y24=1 at A and D and meets the coordinate axis at B and C such that PA⋅PD=PB⋅PC, then range of λ is (correct answer + 2, wrong answer - 0.50)

A
(,6][6,)
Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses
B
(,12][12,)
No worries! We‘ve got your back. Try BYJU‘S free classes today!
C
[6,)
No worries! We‘ve got your back. Try BYJU‘S free classes today!
D
[12,)
No worries! We‘ve got your back. Try BYJU‘S free classes today!
Open in App
Solution

## The correct option is A (−∞,−6]∪[6,∞)Given : x29+y24=1 Any point on the line passing through P(λ,2) is (λ+rcosθ,2+rsinθ) Intersection with ellipse, we get (λ+rcosθ)29+(2+rsinθ)24=1⇒(4cos2θ+9sin2θ)r2+(8λcosθ+36sinθ)r+4λ2=0⇒(5sin2θ+4)r2+(8λcosθ+36sinθ)r+4λ2=0 The roots are r1,r2, so r1r2=4λ25sin2θ+4⇒PA⋅PD=4λ25sin2θ+4 Intersection with x− axis 2+r3sinθ=0⇒r3=−2sinθ Intersection with y− axis λ+r4cosθ=0⇒r4=−λcosθ⇒PB⋅PC=r3r4=2λsinθcosθ PA⋅PD=PB⋅PC⇒4λ25sin2θ+4=2λsinθcosθ⇒λsin2θ=5(1−cos2θ)2+4⇒2λsin2θ+5cos2θ=13⇒−1≤13√4λ2+25≤1⇒0≤1694λ2+25≤1⇒144≤4λ2⇒λ2≥36∴λ∈(−∞,−6]∪[6,∞)

Suggest Corrections
0
Join BYJU'S Learning Program
Related Videos
Ellipse and Terminologies
MATHEMATICS
Watch in App
Explore more
Join BYJU'S Learning Program