The equation |z-z1|^2+|z-z2|^2=k, k ∈R represents a circle if (A) k ≥(1)/(2)|z1-z2|^2 (B) k ≤(1)

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The equation |z-z1|^2+|z-z2|^2=k, k ∈R represents a circle...

The equation $\left|z-z_{1}\right|^{2}+\left|z-z_{2}\right|^{2}=k, k \in R$ represents a circle if (A) $k \geq \frac{1}{2}\left|z_{1}-z_{2}\right|^{2}$ (B) $k \leq \frac{1}{2}\left|z_{1}-z_{2}\right|^{2}$ (C) $k \geq \frac{1}{2}\left|z_{1}+z_{2}\right|^{2}$ (D) $k \leq \frac{1}{2}\left|z_{1}+z_{2}\right|^{2}$

The equation $\left|z-z_{1}\right|^{2}+\left|z-z_{2}\right|^{2}=k, k \in R$ represents a circle if
(A) $k \geq \frac{1}{2}\left|z_{1}-z_{2}\right|^{2}$
(B) $k \leq \frac{1}{2}\left|z_{1}-z_{2}\right|^{2}$
(C) $k \geq \frac{1}{2}\left|z_{1}+z_{2}\right|^{2}$
(D) $k \leq \frac{1}{2}\left|z_{1}+z_{2}\right|^{2}$

Key Concepts

Condition for the Existence of a Real Circle

For the equation representing a circle to be meaningful, its radius must be a real, non-negative quantity. This imposes a constraint on the constant in the equation relative to the positions of the fixed points (as given by the squared distance between them). Specifically, ensuring that the constant is greater than or equal to a certain threshold (in this context, related to half the squared distance between the two fixed points) guarantees that the equation defines a real circle.

Complex Numbers and Their Moduli

This concept involves understanding complex numbers and the modulus operation, which represents the distance of a complex number from the origin in the complex plane. The modulus is essential when expressing geometrical loci, like circles, since it provides a direct measure of distance using the absolute value of differences between complex numbers.

Representation of Circles in the Complex Plane

Circles in the complex plane can be represented using equations that involve the distances between an arbitrary point z and fixed points. Such equations typically take the form of modulus expressions, and the standard form of a circle’s equation involves specifying a center and a radius. Recognizing how to manipulate and rewrite these expressions is key to identifying the type of geometric figure described.

Geometric Interpretation of Sum of Squared Distances

The equation given, involving the sum of the squares of distances from an arbitrary point to two fixed points, can be interpreted geometrically. By completing the square or using algebraic manipulations, one can show that the locus of points satisfying such an equation is a circle (or sometimes a degenerate case), provided a certain condition on the constant term is met. This process demonstrates how algebraic conditions directly tie into geometric shapes.

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