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The line

y = m x + c

is a tangent to the circle with equation

x^{2} + y^{2} = r^{2}

Let the point of tangency be $P .$

Find, in terms of $m$ and $c,$ the coordinates of $P .$
Hence, prove that the tangent line is perpendicular to the radius joining the origin to $P .$

To find the coordinates of $P,$ substitute the line $y = m x + c$ into the equation of the circle

x^{2} + y^{2} = r^{2}

You will get a quadratic equation in terms of $x .$ Use the formula.

Remember that, if there is only one intersection, then $b^{2} - 4 a c = 0$ . This makes the equation much simpler.

If you get the coordinates of $P$ correct, then the gradient of the radius $O P$ will become something very simple and familiar.