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  4. Equation of a tangent to the hyperbola 5x2 - y2 = 5 and which passes through an external point (2, 8) is

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Q. Equation of a tangent to the hyperbola $5x^2 - y^2 = 5$ and which passes through an external point (2, 8) is

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Solution:

Let the tangent be $\quad y = mx \pm \sqrt{m^{2}-5}$
Since it passes through (2, 8) $\quad\Rightarrow \,\left(8-2m\right)^{2} = m^{2}-5$
$\Rightarrow 3m^{2} - 32m + 69 = 0\quad \Rightarrow 3m^{2} - 9m - 23m + 69 = 0 \quad\Rightarrow \left(3m - 23\right) \left(m - 3\right) = 0\quad \Rightarrow m = 3$ or $\frac{23}{3}$