Trigonometry and equation of a circle

Question

Consider the circle with equation x^2+y^2-4y=0 and the point P(5,2). Draw a diagram to show the circle and the two lines from P that are tangent to the circle. Find the angle between the two tangent lines, angle APB, where A and B are the two points of contact.

Answer 1

Answer 2

Use completing the square to put the equation of the circle into standard form:

x² + y² - 4y = 0 

x² + y² - 4y + 4 = 4

x² + (y - 2)² = 4

Circle center is (0,2) with radius of 2  

sin(θ) = 2/5

θ = sin^-1(2/5) = 23.58°

The angle between the 2 lines is 2θ = 47.16°