Let the line y = mx + c be a tangent to the circle x2 + y2 = a2 at (x1, y1)
The equation of the tangent at (x1, y1) to the circle x2 + y2 = a2 is xx1 + yy1 = a2
Thus, the equations y = mx + c and xx1 + yy1 = a2 are representing the same straight line and hence their coefficients are proportional.
1/y1 = −m/x1 = c/a2
x1 = −a2m/c
y1 = a2/c
(x1, y1) is a point on the circle x2 + y2 = a2
∴ x12 + y12 = a2
Put the value of x1 and y1, you will get the required condition.
c2 = a2(1 + m2) is the required condition.
The point of contact of the tangent y = mx + c to the circle x2 + y2 = a2 is