# Complex Numbers – Position of a Point with respect to Circle

## Complex Numbers – Position of a Point with respect to Circle

Position of a Point with respect to Circle: Let the general equation of the circle is $$z\bar{z}+a\bar{z}+\bar{a}z+b=0$$. Whose center z₀ and radius a. there are following position with respect to circle are given below.

a) If the point (z₀) lies inside the circle, then |z – z₀| < a.

b) If the point (z₀) lies on the circle, then |z – z₀| = a.

c) If the point (z₀) lies outside the circle, then |z – z₀| > a.

Nature of Circle: The general equation of circle $$z\bar{z}+a\bar{z}+\bar{a}z+b=0$$ represents a

• Real circle, if |a|² > b.
• Point circle, if |a|² = b.
• An imaginary circle, if |a|² < b.

Problem: If z lies on the circle |z-1|=1, then (z – 2)/ z is equal to.

Solution: Given that,

z lies on the circle |z – 1|=1

|z – 1| = 1 represent a circle whose diameter is a line segment joining the point z = 0 and z = 2

If z lies on a circle, then (z – 2)/ (z – 0).

I.e., (z – 2)/ z is purely imaginary.