SATHEE: Hyperbola 1 Question 17

Hyperbola 1 Question 17

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

	List-I	List-II
P.	The length of the conjugate axis of $H$ is	1.	8
Q.	The eccentricity of $H$ is	2.	$\frac{4}{\sqrt{3}}$
R.	The distance between the foci of $H$ is	3.	$\frac{2}{\sqrt{3}}$
S.
The length of the latus rectum of $H$ is	4.

The correct option is

(2018 Adv.)

(a) $P \to 4; Q \to 2; R \to 1; S \to 3$

(b) $P \to 4; Q \to 3; R \to 1; S \to 2$

(d) $P \to 3; Q \to 4; R \to 2; S \to 1$

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Solution:

We have,

Equation of hyperbola

It is given,

$∠ L N M = 60^{\circ}$

and Area of $△ L M N = 4 \sqrt{3}$

Now, $△ L N M$ is an equilateral triangle whose sides is $2 b$

$[∵ △ L O N ≅ △ M O L; ∴ ∠ N L O = ∠ N M O = 60^{\circ}]$

$∴$ Area of $△ L M N = \frac{\sqrt{3}}{4} (2 b)^{2}$

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

Also, area of $△ L M N = \frac{1}{2} a (2 b) = a b$

$\Rightarrow 4 \sqrt{3} = a (2) \Rightarrow a = 2 \sqrt{3}$

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

(Q) Eccentricity $(e) = \sqrt{1 + \frac{b^{2}}{a^{2}}} = \sqrt{1 + \frac{4}{12}} = \frac{4}{2 \sqrt{3}} = \frac{2}{\sqrt{3}}$

(R) Distance between the foci $= 2 a e = 2 \times 2 \sqrt{3} \times \frac{2}{\sqrt{3}} = 8$

(S) The length of latusrectum $= \frac{2 b^{2}}{a} = \frac{2 (4)}{2 \sqrt{3}} = \frac{4}{\sqrt{3}}$

$P \to 4; Q \to 3; R \to 1; S \to 2$

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Hyperbola 1 Question 17

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

Solution:

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Hyperbola 1 Question 17

18. Let H:x2a2−y2b2=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.

Solution:

इंजीनियरिंग की तैयारी

चिकित्सा तैयारी

एनसीईआरटी पुस्तकें और समाधान

महत्वपूर्ण संसाधन

अन्य संसाधन

पाठ्यक्रम

परीक्षा मंच

नमूना मॉक टेस्ट

सदस्यता

एनसीईआरटी व्यायाम वीडियो समाधान

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