Find the condition that the line $y = m x + c$ is tangent to the circle $x^{2} + y^{2} = a^{2}$

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Solution

The equation of the line is
$y = m x + c$
$\Rightarrow$ $m x - y + c = 0$ .......(i)
Here, $a = m, b = - 1, c = c$
The equation of the circle is
$x^{2} + y^{2} = a^{2}$ .....(ii)
Line (i) will be a tangent to the circle (ii), if the length of perpendicular from the centre $(0, 0)$ of circle (ii) on line (i) $=$ the radius $a$ .
$\Rightarrow \pm \frac{m (0) - 0 + c}{\sqrt{m^{2} + 1}} = a$
$\Rightarrow c = \pm a \sqrt{(1 + m^{2})}$
$\Rightarrow c^{2} = a^{2} {(1 + m}^{2})$
Which is the required condition.

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