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  • The foci of a hyperbola are and its eccentricity is <img alt="\frac{3}{2}" src="https://entranc

The foci of a hyperbola are ( \pm 2,0) and its eccentricity is \frac{3}{2}. A tangent, perpendicular to the line 2 x+3 y=6, is drawn at a point in the first quadrant on the hyperbola. If the intercepts made by the tangent on the \mathrm{x} and \mathrm{y} axes are \mathrm{a} and \mathrm{b} respectively, then |6 \mathrm{a}|+|5 \mathrm{~b}|is equal to ______.

Option: 1

12


Option: 2

-


Option: 3

-


Option: 4

-


Answers (1)

best_answer

\begin{aligned} & 2 \mathrm{ae}=4 \\ & 2 \mathrm{a}\left(\frac{3}{2}\right)=4 \end{aligned}

$$ \begin{aligned} & \Rightarrow a=\frac{4}{3} \\ & e^2=1+\frac{b^2}{a^2} \\ & \Rightarrow \frac{9}{4}=1+b^2\left(\frac{9}{16}\right) \\ & \Rightarrow b^2=\left(\frac{5}{4}\right)\left(\frac{16}{9}\right)=\frac{20}{9} \end{aligned}\\ $$ slope of tangent m=\frac{3}{2}

equation of tangent is

\begin{aligned} & y=m x \pm \sqrt{a^2 m^2-b^2} \\ & y=\frac{3}{2} x \pm \sqrt{\frac{16}{9}\left(\frac{9}{4}\right)-\frac{20}{9}} \\ & \Rightarrow y=\frac{3 x}{2} \pm \frac{4}{3} \\ & y=0 \Rightarrow a= \pm \frac{8}{9} \\ & x=0 \Rightarrow b= \pm \frac{4}{3} \\ & |6 a|+|5 b|=\frac{16}{3}+\frac{20}{3}=12 \end{aligned}

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Nehul

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