Hyperbola 1 Question 17

18.

Let H:x2a2y2b2=1, where a>b>0, be a hyperbola in the XY-plane whose conjugate axis LM subtends an angle of 60 at one of its vertices N. Let the area of the LMN be 43.

List-I List-II
P. The length of the
conjugate axis of H is
1. 8
Q. The eccentricity of H is 2. 43
R. The distance between the
foci of H is
3. 23
S. The length of the latus
rectum of H is
4. 4

The correct option is

(a) P4;Q2;R1;S3

(b) P4;Q3;R1;S2

(c) P4;Q1;R3;S2

(d) P3;Q4;R2;S1

(2018 Adv.)

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Answer:

Correct Answer: 18. (b)

Solution:

  1. We have,

Equation of hyperbola

It is given,

LNM=60

and Area of LMN=43

Now, LNM is an equilateral triangle whose sides is 2b

LONMOL;

NLO=NMO=60

Area of LMN=34(2b)2

43=3b2b=2

Also, area of LMN=12a(2b)=ab

43=a(2)a=23

(P) Length of conjugate axis =2b=2(2)=4

(Q) Eccentricity (e)=1+b2a2=1+412=423=23

(R) Distance between the foci =2ae=2×23×23=8

(S) The length of latusrectum =2b2a=2(4)23=43

P4;Q3;R1;S2



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