Hyperbola (JEE Advanced)

Question 1

Let S and S' be the foci of (x²)/16−(y²)/9=1 and let P be a point on it with PS'=3 PS. The area of triangle PSS' is ______. Exactly one option is correct.

Accepted Answer

C. Here a=4, b=3, c=5, SS'=10. For any point of the hyperbola, lvert PS'−PSrvert=2a=8.
With PS'=3PS: 2PS=8, so PS=4, PS'=12. (The order PS'=3PS means P is nearer S, i.e. on the branch of S; e.g. S=(5,0), PS=5/4x−4=4 gives x=32/5≥4, a genuine point.)
Sides 12,4,10: s=13, area =√(13·1·9·3)=√351=3√39.
Check: y²=9(((32/5)²)/16−1)=1404/100, and 1/2·10·lvert yrvert=5√14.04=3√39.
Difficulty: Hard

Question 2

Consider a hyperbola whose foci subtend a given angle at an end of its conjugate axis. The hyperbola is (x²)/(a²)−(y²)/(b²)=1 (a,b>0), with transverse axis along the x-axis (length 2a), conjugate axis

Accepted Answer

A. By symmetry BO bisects ∠ SBS', so ∠ OBS=π/3 and c/b=tanπ/3=√3, i.e. c²=3b².
Then a²=c²−b²=2b², so b/a=1/(√2) (and e²=3/2).
The asymptotes y=±b/ax have slopes ±1/(√2). With tanα=1/(√2), the angle between them is 2α where tan2α=(2/√2)/(1−1/2)=2√2>0, so 2α=tan⁻¹(2√2)≈70.5° is already acute.
Difficulty: Hard

Question 3

The hyperbola (x²)/4−(y²)/5=1 and the parabola y²=12x have common tangents. The length of the segment of any one common tangent between its points of contact with the two curves is ______. Exactly one

Accepted Answer

D. A non-vertical tangent to y²=12x is y=mx+3/m (m≠0); the only vertical tangent x=0 does not touch the hyperbola.
y=mx+k touches (x²)/4−(y²)/5=1 iff k²=4m²−5 (with k≠0). So 9/(m²)=4m²−5, i.e. 4m⁴−5m²−9=0, (4m²−9)(m²+1)=0, m=±3/2.
Take m=3/2: the tangent is y=3/2x+2.
Contact with the parabola: (3/(m²),6/m)=(4/3,4).
Contact with the hyperbola: (−(a²m)/k,−(b²)/k)=(−3,−5/2); check 9/4−(25/4)/5=1 (left b

Hyperbola: questions with solutions

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