Let H:x2a2 -y2b2=1 , where a>b>0, be a hyperbola in the xy-plane whose conjugate axis LM subtends an angle of 60o at one of its vertices N. Let the area of the triangle LMN be 43. LIST-I LIST-II A. The length of the conjugate axis of H is P. 8 B. The eccentricity of H is Q. 43 C. The distance between the foci of H is R. 23 D. The length of the latus rectum of H is S. 4 The correct option is:

JEE Advanced · Mathematics · 15. Hyperbola

Let $H : \frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$ , where $a > b > 0$ , be a hyperbola in the xy-plane whose conjugate axis $L M$ subtends an angle of $60^{o}$ at one of its vertices $N$ . Let the area of the triangle $L M N$ be $4 \sqrt{3}$ .

LIST-I LIST-II

A. The length of the conjugate axis of $H$ is P. $8$

B. The eccentricity of $H$ is Q. $\frac{4}{\sqrt{3}}$

C. The distance between the foci of $H$ is R. $\frac{2}{\sqrt{3}}$

D. The length of the latus rectum of $H$ is S. $4$

The correct option is:

A a-r;b-q;c-s;d-p;
B a-q;b-r;c-s;d-p;
C a-r;b-s;c-p;d-q;
D a-s;b-r;c-p;d-q;

Verified Solution

Answer & Solution

Correct Answer

(D) a-s;b-r;c-p;d-q;

Step-by-step Solution

Detailed explanation

\tan 30^{o} = \frac{b}{a}

\Rightarrow

a = b \sqrt{3}

Now area of

∆ L M N = \frac{1}{2} . 2 b . b \sqrt{3}

4 \sqrt{3} = \sqrt{3} b^{2}

\Rightarrow b = 2 & a = 2 \sqrt{3}

\Rightarrow e = \sqrt{1 + \frac{b^{2}}{a^{2}}} = \frac{2}{\sqrt{3}}

P. Length of conjugate axis

= 2 b = 4

P \to 4

Q. Eccentricity

e = \frac{2}{\sqrt{3}}

Q \to 3

R. Distance between foci

= 2 a e

= 2 (2 \sqrt{3}) (\frac{2}{\sqrt{3}}) = 8

So,

R \to 1

S. Length of latus rectum

= \frac{2 b^{2}}{a} = \frac{2 {(2)}^{2}}{2 \sqrt{3}} = \frac{4}{\sqrt{3}}

S \to 2

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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Match List I with List II and select the correct answer using the codes given below the lists:

	$\text{List I}$		$\text{List II}$
a.	Boltzmann constant	p.	$\left[M L^2 T^{-1}\right]$
b.	Coefficient of viscosity	q.	$\left[M L^{-1} T^{-1}\right]$
c.	Planck constant	r.	$\left[M L T^{-3} K^{-1}\right]$
d.	Thermal conductivity	s.	$\left[M L^2 T^{-2} K^{-1}\right]$

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LIST-I	LIST-II
A. The length of the conjugate axis of $H$ is	P. $8$
B. The eccentricity of $H$ is	Q. $\frac{4}{\sqrt{3}}$
C. The distance between the foci of $H$ is	R. $\frac{2}{\sqrt{3}}$
D. The length of the latus rectum of $H$ is	S. $4$

	\(\text{List I}\)		\(\text{List II}\)
a.	Boltzmann constant	p.	\(\left[M L^2 T^{-1}\right]\)
b.	Coefficient of viscosity	q.	\(\left[M L^{-1} T^{-1}\right]\)
c.	Planck constant	r.	\(\left[M L T^{-3} K^{-1}\right]\)
d.	Thermal conductivity	s.	\(\left[M L^2 T^{-2} K^{-1}\right]\)