Problem 102
Question
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) \(k \leq \frac{1}{2}\left|z_{1}+z_{2}\right|^{2}\)
Step-by-Step Solution
Verified Answer
(A) \(k \geq \frac{1}{2}|z_1 - z_2|^2\) ensures the equation represents a circle.
1Step 1: Understand the Equation
The given equation is \(\left|z-z_{1}\right|^{2} + \left|z-z_{2}\right|^{2} = k\). We need to show under what condition this equation represents a circle. \(|z-z_1|\) and \(|z-z_2|\) represent the distances of a point \(z\) from two fixed points \(z_1\) and \(z_2\) in the complex plane.
2Step 2: Substitute and Expand Distances
Express \(|z-z_1|^2\) and \(|z-z_2|^2\) in terms of the components of \(z = x + yi\), \(z_1 = x_1 + y_1i\), and \(z_2 = x_2 + y_2i\). Expanding, \(|z-z_1|^2 = (x-x_1)^2 + (y-y_1)^2\) and \(|z-z_2|^2 = (x-x_2)^2 + (y-y_2)^2\).
3Step 3: Simplify the Equation
Combine and rearrange these expansions to the form \((x-x_0)^2 + (y-y_0)^2 = r^2\), where \((x_0, y_0)\) is the center and \(r\) is the radius. Doing this requires combining like terms and factoring.
4Step 4: Identify the Minimum Check
Realize that the minimum value of \(|z-z_1|^2 + |z-z_2|^2\) occurs geometrically when \(z\) is the midpoint of \(z_1\) and \(z_2\). In such case, the expression minimizes to \(\left|\frac{z_1 - z_2}{2}\right|^2\).
5Step 5: Determine the Condition for a Circle
For the expression to represent a circle, \(k\) must be greater than or equal to twice this minimum value, i.e., \(k \geq \frac{1}{2}|z_1 - z_2|^2\), as this ensures a non-zero radius.
Key Concepts
Geometric RepresentationEquation of a CircleComplex Plane
Geometric Representation
Complex numbers, though often appearing as abstract constructs, can be understood quite visually using geometric representation. Consider a complex number \( z = x + yi \), where \( x \) and \( y \) are real numbers. This can be plotted on the complex plane, a two-dimensional plane where the horizontal axis represents the real part \( x \) and the vertical axis represents the imaginary part \( y \).
In this plane, any complex number corresponds to a unique point. The geometric power of complex numbers comes into play when considering transformations, like rotations or translations. Specifically, the modulus \(|z-z_1|\) can be visualized as the Euclidean distance from \( z \) to a fixed point \( z_1 \).
Thus, the expression \( |z-z_1|^2 + |z-z_2|^2 = k \) involves distances from \( z \) to two fixed points \( z_1 \) and \( z_2 \). Geometrically, this equation suggests the locus of all points whose total distance to these fixed points remains constant. This is a fascinating concept because it eventually leads us to the definition of a geometric shape - typically a circle, as explored further in our later sections.
In this plane, any complex number corresponds to a unique point. The geometric power of complex numbers comes into play when considering transformations, like rotations or translations. Specifically, the modulus \(|z-z_1|\) can be visualized as the Euclidean distance from \( z \) to a fixed point \( z_1 \).
Thus, the expression \( |z-z_1|^2 + |z-z_2|^2 = k \) involves distances from \( z \) to two fixed points \( z_1 \) and \( z_2 \). Geometrically, this equation suggests the locus of all points whose total distance to these fixed points remains constant. This is a fascinating concept because it eventually leads us to the definition of a geometric shape - typically a circle, as explored further in our later sections.
Equation of a Circle
The equation of a circle can be derived from basic geometric principles, even when considering complex numbers in the complex plane. If you recall the standard form of a circle's equation in the Cartesian coordinate system, it is \((x-h)^2 + (y-k)^2 = r^2\), where \((h, k)\) is the center and \(r\) is the radius.
In the context of the complex plane, for our given exercise, \(|z-z_1|^2 + |z-z_2|^2 = k\) can be expanded and rearranged into a similar form: \[(x - x_0)^2 + (y - y_0)^2 = r^2\]. This reveals that the equation indeed represents a circle.
Although complex, the exercise simplifies to understanding that, for any \(z\), maintaining a particular sum of squares of distances implies circular symmetry. Achieving this requires ensuring \( k \) is equal to or larger than a certain minimum threshold, based on the properties of distances in the complex plane. Particularly, as demonstrated in the solution, the criterion \( k \geq \frac{1}{2}|z_1 - z_2|^2 \) ensures the expression covers a valid circle with a nonzero radius.
In the context of the complex plane, for our given exercise, \(|z-z_1|^2 + |z-z_2|^2 = k\) can be expanded and rearranged into a similar form: \[(x - x_0)^2 + (y - y_0)^2 = r^2\]. This reveals that the equation indeed represents a circle.
Although complex, the exercise simplifies to understanding that, for any \(z\), maintaining a particular sum of squares of distances implies circular symmetry. Achieving this requires ensuring \( k \) is equal to or larger than a certain minimum threshold, based on the properties of distances in the complex plane. Particularly, as demonstrated in the solution, the criterion \( k \geq \frac{1}{2}|z_1 - z_2|^2 \) ensures the expression covers a valid circle with a nonzero radius.
Complex Plane
The complex plane serves as a valuable tool in visualizing complex numbers and their interactions. It is a fundamental two-dimensional plane where any complex number \( z = x + yi \) is represented as a point. Here, the real part \( x \) is on the horizontal axis, and the imaginary part \( y \) is on the vertical axis.
Working within the complex plane allows us to interpret mathematical problems geometrically. For instance, distances and transformations can be directly observed, offering intuitive understanding. This is crucial when analyzing equations like \( |z - z_1|^2 + |z - z_2|^2 = k \).
Working within the complex plane allows us to interpret mathematical problems geometrically. For instance, distances and transformations can be directly observed, offering intuitive understanding. This is crucial when analyzing equations like \( |z - z_1|^2 + |z - z_2|^2 = k \).
- Distance: The modulus \(|z - z_1|\) describes the distance between two points: \(z\) and \(z_1\).
- Geometry: The equation represents a locus of points forming a circle based on constant total distance.
Other exercises in this chapter
Problem 99
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View solution Problem 100
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