Let H : x 2 a 2   - 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 3 . LIST-I LIST-II A. The length of the conjugate axis of H is P. 8 B. The…

Question

Let H : x 2 a 2 &#160; - y 2 b 2 = 1 , where a &#62; b &#62; 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 3 . LIST-I LIST-II A. The length of the conjugate axis of H is P. 8 B. The eccentricity of H is Q. 4 3 C. The distance between the foci of H is R. 2 3 D. The length of the latus rectum of H is S. 4 The correct option is:

Aicharya · Accepted Answer

tan ⁡ 30 o = b a ⇒ a = b 3 Now area of ∆ L M N = 1 2 . 2 b . b 3 4 3 = 3 b 2 ⇒ b = 2 & a = 2 3 ⇒ e = 1 + b 2 a 2 = 2 3 P. Length of conjugate axis = 2 b = 4 So P → 4 Q. Eccentricity e = 2 3 So Q → 3 R. Distance between foci = 2 a e = 2 2 3 2 3 = 8 So, R → 1 S. Length of latus rectum = 2 b 2 a = 2 2 2 2 3 = 4 3 So S → 2

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. $4$

Let H : x 2 a 2   - y 2 b 2 = 1 , where a > b > 0 , be a hyperbola in the xy-plane whose…

Solution

Let H : x 2 a 2 &#160; - y 2 b 2 = 1 , where a &#62; b &#62; 0 , be a hyperbola in the xy-plane whose…

Solution

Let H : x 2 a 2 - y 2 b 2 = 1 , where a > b > 0 , be a hyperbola in the xy-plane whose…