For z and 1 , if - 1 is the radius of the circle | z - | 2 + | z - | 2 = 2 , then is equal to _________. [2023]

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Solution

Q.
$For α, β, z \in ℂ and λ > 1,$ $if \sqrt{λ - 1}$ is the radius of the circle $| z - α |^{2} + | z - β |^{2} = 2 λ,$ then $| α - β |$ is equal to _________. [2023]

Ans.
(2)

For a circle, we have, $| z - z_{1} |^{2} + | z - z_{2} |^{2} = | z_{1} - z_{2} |^{2}$

$r = \frac{| z_{1} - z_{2} |}{2} = \frac{| α - β |}{2} = \sqrt{λ - 1}$

$Also, 2 λ = | α - β |^{2}$

$\Rightarrow 2 λ = 4 (λ - 1) \Rightarrow 2 λ = 4 λ - 4 \Rightarrow 2 λ = 4 \Rightarrow λ = 2$

$∴$ $| α - β |$ $= 2$

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