Find point on circle's tangent based on point on circle, radius and angle

Question

The circle is centered at (0,0)"P" with a radius of 5. I have a point on the circle at (4,-3)"A".

How would I find the points "B1" and "B2" on the tangent through point "A" given an arbitrary angle (A P Bx)?

Answer 1

You can obtain the equation of the tangent at $A$ which is $3y=4x-25$

If the given angle $APB$ is $\theta$ then $PB=5\sec\theta$, so you can form and solve a quadratic equation like $$x^2+(\frac{4x-25}{3})^2=25\sec^2\theta$$