Equation of a tangent passing through $(2, 8)$ to the hyperbola $5 x^{2} - y^{2} = 5$ is :

3 x - y + 2 = 0

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3 x + y + 14 = 0

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23 x - 3 y - 22 = 0

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3 x - 23 y + 178 = 0

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Solution

The correct options are
B $23 x - 3 y - 22 = 0$
C $3 x - y + 2 = 0$
Let the equation of the tangent be
$y = m x + c$
Since it passes through $(2, 8)$ hence
$8 = 2 m + c$
Or
$c = 8 - 2 m$
Hence the equation of the tangent becomes
$y = m x - 2 m + 8$
Substituting in the equation of the hyperbola
$5 x^{2} - y^{2} = 5$
$5 x^{2} - (m x - 2 m + 8)^{2} = 5$
$5 x^{2} - 5 - (m^{2} x^{2} + 4 m^{2} + 64 - 4 m^{2} x - 32 m + 16 m x) = 0$
$x^{2} (5 - m^{2}) + x (4 m^{2} - 16 m) - 4 m^{2} + 32 m - 69 = 0$
Since it is a tangent hence D is 0.
Or
$B^{2} - 4 A C = 0$
$(4 m^{2} - 16 m)^{2} - 4 (5 - m^{2}) (- 4 m^{2} + 32 m - 69) = 0$
$m = 3$ and $m = \frac{23}{3}$
Hence
$y = 3 x + 2$ and $3 y = 23 x - 22$

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