JEE Main 2022 · 28 Jun · Shift 2 · Mathematics Q13

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MathematicsSingle correct+4 / -1Question 13 of 90

Let $a>0$ , $b>0$ . Let $e$ and $l$ respectively be the eccentricity and length of the latus rectum of the hyperbola $\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1$ . Let $e'$ and $l'$ respectively be the eccentricity and length of the latus rectum of its conjugate hyperbola. If $e^2=\dfrac{11}{14}l$ and $\left(e'\right)^2=\dfrac{11}8l'$ , then the value of $77a+44b$ is equal to :

100

110

120

130

Answer and worked solution

Official answer: D

Conditions. $\frac{a^2+b^2}{a^2}=\frac{11}{14}\cdot\frac{2b^2}a$ and $\frac{a^2+b^2}{b^2}=\frac{11}8\cdot\frac{2a^2}b$ , i.e. $7(a^2+b^2)=11ab^2$ and $4(a^2+b^2)=11a^2b$ .

Divide. $\frac74=\frac ba$ ; then $a=\frac{65}{77}$ , $b=\frac{65}{44}$ .

Value. $77a+44b=130$ .

(D)

Topics:Hyperbola

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JEE Main 2022 · 28 Jun · Shift 2, Mathematics question 13