Mathematics

Problem

Let

S(x,y)=x^2+y^2+2gx+2fy+c

represent a circle and let

A(x_1,y_1)

and

B(x_2,y_2)

be two points in the plane such that

S_1<0

for

A

and

S_2>0

for

B

. a. (i) Prove that the product

S_1S_2

is negative. b. (ii) Show that the straight line segment joining

A

and

B

must intersect the circle at least once. c. (iii) If

S_1=0

S_2=0

, explain how your conclusion in part (ii) adapts to these boundary cases.