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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) is  O

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}
Also, \mathrm{c^2=\left(1+\frac{b^2}{a^2}\right) \Rightarrow 2^2-1=\frac{b^2}{a^2}}

[Given, eccentricity =2 ]

\mathrm{\Rightarrow b^2=3 a^2}

Since, hyperbola passes through the point (4,6).

\begin{aligned} &\mathrm{ \therefore \frac{16}{a^2}-\frac{36}{b^2}=1 \Rightarrow \frac{16 \times 3-36}{3 a^2}=1\left[\because b^2=3 a^2\right] }\\ &\mathrm{ \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-4}{4}-\frac{y \cdot 6}{12}=1 \Rightarrow x-\frac{y}{2}=1 \Rightarrow 2 x-y-2=0 }

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Pankaj

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