# Tangent to the circle given a point it must pass through a point on another circle

I need to find the angle at origin caused by two lines (one is the radius of one circle, and the other is tangent to the other circle).

The Point A on the green circle is known, the dashed line and shorter full line begin at the origin. The other line is tangent to the red circle and passing through Point A.

How would I calculate the angle a?

I know this is probably a basic question but my head is burnt out at work and I just need reassurance before making an epic mistake.

• The radius of the big circle is $\sqrt{2}$ but the radius $r$ of a small circle is unknown. The angle $\alpha$ depends on $r.$ Mar 13, 2014 at 10:24
• The radius of the red circle...and also: are both circles centered at the origin? Mar 13, 2014 at 10:50

Since from the drawing it is clear that $\;r<\sqrt 2\;$, we have that if $\;\alpha\;$ is the wanted angle then
$$\cos\alpha=\frac r{\sqrt2}\implies \alpha=\arccos\frac r{\sqrt2}$$
• In fact the point $\;(1,1)\;$ has no importance whatsoever in this case, @Amie-lea : going by the same drawing, any other point on the outer circle gives the very same result. Mar 13, 2014 at 11:17