Prove: Any line L will cut a circle **at most 2** points:

Let circle C (x,y) be unit circle defined by

C(x,y) : x² + y² = 1

Factorize C(x,y) : (x+iy) (x-iy) = 1 in the complex plane.

So C = {L1} U {L2}

where L1 and L2 are two lines

L1= x+iy

L2= x – iy

L1 and L2 intersect at origin (0,0):

x+ iy = x-iy

We know that any line L will cut L1

**at most**1 point, and L2**at most**1 pointTherefore,

L cuts the circle C

**at most**(1+1=) 2 points. [QED]