Need to find several pieces of information given partial information about 2 circles.

Question

I've got 2 circles, Circle 1 with radius $R_1$, and Circle 2 with radius $R_2$. Circle 1 is inscribed within Circle 2 and $R_1$ will always be less than $R_2$. The center point of Circle 1 will always be $(R_1, 0)$. The center point of circle 2 will always be on the $Y$-axis $(0, Y_2)$. The two circles intersect at one and only one point.

I need to find the following:

1. The value of $Y_2$ (the center point of circle 2).
2. The intersection point of Circle 1 & Circle 2.
3. The slope of the tangent line to both circles at that point.
4. The point on Circle 1 where a line tangent to Circle 1 has a slope of $-110 (70)$ degrees.
5. The point on Circle 2 where a line tangent to Circle 2 has a slope of $-165 (15)$ degrees.

For example, in this diagram $R_1 = 6$, $R_2 = 21$.

Answer 1

Hint: draw a circle with the radius $R_2-R_1$ and center at $(R_1,0)$. The intersection of that circle with the axis $y$ will be the center of the larger circle. You will see that there will be two possible positions for the center of the larger circle: $(0,\sqrt{R_2^2-2R_1R_2})$ and $(0,-\sqrt{R_2^2-2R_1R_2})$.

Answer 2

In your figure $AD=R_1$, $CD=R_2$. Then $\triangle ABC$ is a right angle triangle. One of the sides is $AB=R_1$, the hypotenuse is $AC=R_2-R_1$. Use Pythagoras' theorem to find $BC$. Once you have it, you can find all other answers in your problem.