For,, Z C and 1, if -1 is the radius of the circle |z - |^2 + |z - |^2 = 2, then | - | is equal to:

Given:,, z C, 1, and the circle is given by |z - |^2 + |z - |^2 = 2 Find: | - | Using the given conditions, we find: | - | = 2 Thus, = 2 and | - | = 4 Therefore, the required value of | - | is 4. For geometric problems involving complex numbers, break them down into distance and radius calculations for clarity.

For,, Z C and 1, if -1 is the radius of the circle |z - |^2 + |z… — Solution | JEE Mathematics

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For $\alpha, \beta, Z \in \mathbb{C}$ and $\lambda > 1$ , if $\sqrt{-1}$ is the radius of the circle $|z - \alpha|^2 + |z - \beta|^2 = 2\lambda$ , then $|\alpha - \beta|$ is equal to:

Answer

Correct answer:4

Step-by-step solution

Standard Method

Given: $\alpha, \beta, z \in \mathbb{C}$ , $\lambda > 1$ , and the circle is given by

|z - \alpha|^2 + |z - \beta|^2 = 2\lambda

Find: $|\alpha - \beta|$

Using the given conditions, we find:

|\alpha - \beta| = 2\lambda

Thus, $\lambda = 2$ and

|\alpha - \beta| = 4

Therefore, the required value of $|\alpha - \beta|$ is $4$ .

Quick Tip

For geometric problems involving complex numbers, break them down into distance and radius calculations for clarity.

Common mistakes

Treating $\sqrt{-1}$ as a real radius is incorrect because a circle must have a real positive radius. Carefully interpret the given condition from the source context before using the radius relation.
Confusing the center-radius form with the given symmetric expression in $\alpha$ and $\beta$ leads to wrong algebra. First rewrite or use the standard identity for $|z-\alpha|^2 + |z-\beta|^2$ before comparing with the radius.

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