Question

Assertion :The equation $z\overline{z}+\overline{b}z+b\overline{z}+\lambda =0$ where b is a complex number representing a circle in Argand plane if $\lambda ϵR.$ Reason: The radius of the circle $z\overline{z}+\overline{b}z+b\overline{z}+\lambda =0$ is $\sqrt{b\overline{b}-\lambda }.$

• A. Both (A) & (R) are individually true & (R) is correct explanation of (A).
• B. Both (A) & (R) are individually true but (R) is not the correct (proper) explanation of (A).
• C. (A) is true (R) is false.
• D. (A) is false (R) is true.

Answer 1

The correct option is A Both (A) & (R) are individually true & (R) is correct explanation of (A).

The given equation can be written as
$b\overline{b}+z\overline{z}+b\overline{z}+b\overline{b}=b\overline{b}-\lambda$
$\Rightarrow (z+b)(\overline{z}+\overline{b})=b\overline{b}-\lambda$
$\Rightarrow |z+b{|}^2=b\overline{b}-\lambda$
$\Rightarrow |z+b|=\sqrt{b\overline{b}-\lambda }$
As $b\overline{b}$ is real, $\lambda$ should be real
$\Rightarrow$ Given equation represent a circle whose radius is $\sqrt{b\overline{b}-\lambda }$
Hence Assertion & Reason both are true & Reason is correct explanation of Assertion.