    Question

# The set of values of α2, if there exists a tangent to the ellipse x2α2+y2=1 such that the portion of the tangent intercepted by the hyperbola α2x2−y2=1 subtends a right angle at the centre of the curves, is

A
[5+12,2]
Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses
B
[1,2]
No worries! We‘ve got your back. Try BYJU‘S free classes today!
C
[512,1]
No worries! We‘ve got your back. Try BYJU‘S free classes today!
D
[512,5+12]
No worries! We‘ve got your back. Try BYJU‘S free classes today!
Open in App
Solution

## The correct option is A [√5+12,2]Equation of any tangent to x2α2+y2=1 in slope form is, y=mx±√α2m2+1 ⇒(y−mx√α2m2+1)2=1 By homogenisation, we can write (α2x2−y2)(α2m2+1)=y2+m2x2−2mxy The portion of tangent subtends 90∘ at the origin. coefficient of x2+coefficient of y2=0 ⇒(α2m2+1)(α2−1)=1+m2 ⇒α2(α2−1)m2+α2−1=1+m2 ⇒[α2(α2−1)−1]m2=2−α2 ⇒m2=2−α2α4−α2−1≥0 ⇒α2−2α4−α2−1≤0 Let α2=t≥0 Then t−2t2−t−1≤0 ⇒t∈[1+√52,2]  Suggest Corrections  0      Similar questions  Explore more