Finding points in circles

Question

so I have two questions Im stuck on and I really do not know what to do at all. Thank you.

1)

2)

Answer 1

For 1) Draw the vertical line through $(x_0,r)$ and the perpendicular to the line at the point of tangency. We get similar triangles, and so:

$$\frac{r}{mx_0 - r} = \frac{x_0}{x_0\sqrt{m^2 + 1}}$$

Cross-multiplying gives: $$r\sqrt{m^2 + 1} = mx_0 - r$$

which simplifies to:

$$x_0 = \frac{r(1 + \sqrt{m^2 + 1})}{m}$$

For 2) Draw the horizontal line at $P_1$, again, we get similar right triangles, so:

$$\frac{r}{a} = \frac{W}{\sqrt{H^2 + W^2}}$$

Cross-multiplying and solving for $a$ gives:

$$a = r\sqrt{1 + \left(\frac{H}{W}\right)^2}$$

Answer 2

For question 1 if you draw lines from the center of the circle perpendicular to the x-axis and from the origin to the center of the circle, you get a right triangle with one leg of length x0 and the other of length r. That means that the tangent of the angle that line from the origin to the center of the circle make is r/x0. And the angle is half the angle, theta, the original line makes so m= tan(theta). You need to use the "half angle" formula for tangent.

Question 2 is similar- use the right triangles.