Question

If $\left( \frac{2 - z_{2}}{3 - z_{2}} \right)$= k, then points A(z₁), B(z₂), C(3, 0) and D(2, 0) taken clockwise –Q

• A. Lie on a circle only if k > 0
• B. Lie on a circle only if k < 0
• C. Lie on a circle " k Ī R
• D. Are vertices of a square " k Ī (0, 1)

Answer 1

Answer: (A)

Lie on a circle only if k > 0

Answer 2

Sol. arg $\left( \frac{3 - z_{1}}{2 - z_{1}} \right)$+ arg $\left( \frac{2 - z_{2}}{3 - z_{2}} \right)$= arg $\left( \frac{3 - z_{1}}{2 - z_{1}}.\frac{2 - z_{2}}{3 - z_{2}} \right)$

Now if k Ī R⁺ then q₁ + q₂ = 0

So q₁ and q₂ are equal in magnitude but opposite in sign.

So DC chord subtends equal angles at A and B.

So points are concyclic for k > 0.