Question

Only question 12 please help.

Answer 1

For 12:
Since the line y = mx + c is a tangent, then the distance from the centre of the circle to the line is equal to the radius.

Using perpendicular distance:
$d = r = \frac{| -m \cdot a + 1 \cdot b - c |}{\sqrt{(-m)^{2} + 1}}$
$r = \frac{|b + am - c|}{\sqrt{m^{2} + 1}}$

Squaring both sides, we get:
$r^{2} = \frac{(b - c + ma)^{2}}{m^{2} + 1}$
$(1 + m^{2})r^{2} = (b - c + ma)^{2}$, as required.