JEE Main202118 Mar 2021Morning ShiftMathematicsComplex NumberActual
If the equation a | z | 2 + α ¯ z + α z ¯ ¯ + d = 0 represents a circle where a , d are real constants then which of the following condition is correct?
Options
- A| α | 2 - a d ≠ 0
- B| α | 2 - a d > 0 and a ∈ R - 0
- C| α | 2 - a d ≥ 0 and a ∈ R
- Dα = 0 , a , d ∈ R +
Correct answer
B. | α | 2 - a d > 0 and a ∈ R - 0
Step-by-step solution
Given a | z | 2 + α ¯ z + α z ¯ ¯ + d = 0 ∵ z 2 = z z ¯   &   α ¯ z + α z ¯ ¯ = α ¯ z ¯ + α z ¯ ¯ = α z ¯ + α ¯ z Then, a z z ¯ + α z ¯ + α ¯ z + d = 0   or z z ¯ + α a z ¯ + α ¯ a z + d a = 0 is the equation of circle. Centre = - α a ,   radius,  r = α α ¯ a 2 - d a = α α ¯ - a d a 2 α a