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  • If the eccentricity of the standard hyperbola passing through the point (4,6) is 2 , then the equation of the tangent to the hyperbola at (4,6) isOptio

If the eccentricity of the standard hyperbola passing through the point (4,6) is 2 , then the equation of the tangent to the hyperbola at (4,6) is

Option: 1

\mathrm{3 x-2 y=0}


Option: 2

\mathrm{x-2 y+8=0}


Option: 3

\mathrm{2 x-y-2=0}


Option: 4

\mathrm{2 x-3 y+10=0}


Answers (1)

best_answer

Standard equation of hyperbola is \mathrm{\frac{x^2}{a^2}-\frac{y^2}{b^2}=1}
\mathrm{ \begin{aligned} & \text { Also, } e^2=\left(1+\frac{b^2}{a^2}\right) \Rightarrow 2^2-1=\frac{b^2}{a^2} \text { [Given, eccentricity = 2] }\, \, \\ & \\ & \Rightarrow b^2=3 a^2 \end{aligned} }
Since, hyperbola passes through the point (4,6).

\mathrm{ \begin{aligned} & \therefore \frac{16}{a^2}-\frac{36}{b^2}=1 \Rightarrow \frac{16 \times 3-36}{3 a^2}=1 \quad\left[\because b^2=3 a^2\right] \\\\ & \Rightarrow 3 a^2=12 \therefore a^2=4 \Rightarrow b^2=12 \end{aligned} }

Now, equation of the tangent to the hyperbola at (4,6) is

\mathrm{ \frac{x \cdot 4}{4}-\frac{y \cdot 6}{12}=1 \Rightarrow x-\frac{y}{2}=1 \Rightarrow 2 x-y-2=0 }

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Divya Prakash Singh

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