Equation of a common tangent with positive slope to the circle as well as to the hyperbola is

3 x - 4 y + 8 = 0

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2 x - \sqrt{5} y + 4 = 0

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4 x - 3 y + 4 = 0

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2 x - \sqrt{5} y - 20 = 0

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Solution

The correct option is B $2 x - \sqrt{5} y + 4 = 0$
Equation of tangent to hyperbola having slope $m$ is
$y = m x + \sqrt{9 m^{2} - 4}$ ...(1)
Equation of tangent to circle is
$y = m (x - 4) + \sqrt{16 m^{2} + 16}$ ...(2)
(1) and (2) will be identical for $m = \frac{2}{\sqrt{5}}$ satisfy
$∴$ Equation of common tangent is $2 x - \sqrt{5} y + 4 = 0$

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Equation of a common tangent with positive slope to the circle as well as to the hyperbola is

The correct option is B 2x−√5y+4=0Equation of tangent to hyperbola having slope m is y=mx+√9m2−4 ...(1) Equation of tangent to circle is y=m(x−4)+√16m2+16 ...(2) (1) and (2) will be identical for m=2√5 satisfy ∴Equation of common tangent is 2x−√5y+4=0