If the eccentricity of the standard hyperbola passing

Question:

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-

  1. $2 x-y-2=0$

  2. $3 x-2 y=0$

  3. $2 x-3 y+10=0$

  4. $x-2 y+8=0$


Correct Option: 1

Solution:

Let us Suppose equation of hyperbola is

$\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$

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

passing through $(4,6) \Rightarrow \mathrm{a}^{2}=4, \mathrm{~b}^{2}=12$

$\Rightarrow$ equaiton of tangent

$x-\frac{y}{2}=1$

$\Rightarrow 2 x-y-2=0$

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