Question

Prove that |z-z1|^2+|z-z2|^2=k will represent a circle

Answer 1

Solution:

We have to prove that the equation

      $|z-z_{1}|^2+|z-z_{2}|^2=k$

                                                       will represent a circle.--------(1)

Let $Z_{1}=a + i b, and Z_{2}= c + id$ be two points in the complex plane.

Supposing , z = x + i y

Writing equation (1) as

(x-a)² +(y-b)²+(x-c)² +(y-d)²=k,→→→ here  |A+ i B|= $\sqrt{A^2 +B^2}$

→→2 x² + 2 y²- 2 x (a +c) - 2 y (b+d) +b²+a²+c²+d²-k=0

Dividing both sides by 2, we get

→→→ x² +  y²-  x (a +c) -  y (b+d) +$\frac{b^2+a^2+c^2+d^2-k}{2}$ =0

Comparing with general equation of circle which is : x² +y²+ 2 g x + 2 f y + c=0

we found that above equation is equation of circle.

Hence,   $|z-z_{1}|^2+|z-z_{2}|^2=k$   represent a circle.