Problem 39 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

Let \(\mathrm{z} \in \mathrm{C}\), the set of complex numbers. Then the equation, \(2|z+3 i|-|z-i|=0\) represents: (a) a circle with radius \(\frac{8}{3}\). (b) a circle with diameter \(\frac{10}{3}\). (c) an ellipse with length of major axis \(\frac{16}{3}\). (d) an ellipse with length of minor axis \(\frac{16}{9}\)

Step-by-Step Solution

Answer

The equation represents an ellipse with a major axis length of \(\frac{16}{3}\).

Step 1: Identify the Given Equation

We start with the equation \(2|z+3i| - |z-i| = 0\). This equation involves complex numbers and modulus, and we're tasked with determining the geometric shape it describes.

Step 2: Rearrange and Simplify the Equation

Rearrange the equation to \(2|z + 3i| = |z - i|\), meaning twice the distance from \(z\) to \(-3i\) is equal to the distance from \(z\) to \(i\).

Step 3: Visualize the Geometric Shape

This equation can be visualized as a locus where for every point \(z\), the sum of distances from two fixed points, known as foci, \(-3i\) and \(i\) maintains a constant ratio. This condition satisfies the definition of an ellipse.

Step 4: Confirm the Shape and Properties

Given an ellipse, the constant ratio for distances from a fixed point to any point on the ellipse determines its geometric properties. Since \(2|z + 3i| = |z - i|\) implies the distances are in the ratio 2:1, this suggests an ellipse rather than a circle. The major axis combines these distances.

Step 5: Calculate the Length of the Major Axis

Let's find the distance between foci: the imaginary distance between \(-3i\) and \(i\) is \( |i - (-3i)| = |i + 3i| = 4 \). Since the equation transforms to an ellipse where \(2d_1 = d_2\), it implies that the sum of distances (major axis length) is twice this, hence \(\frac{16}{3}\).

Key Concepts

Complex Numbers

Complex numbers are an extension of the real number system, adding a dimension to it in the form of imaginary numbers. A complex number is usually expressed as \( z = a + bi \), where \( a \) is the real part and \( bi \) is the imaginary part. Here, \( i \) represents the imaginary unit with the property \( i^2 = -1 \).
It is crucial to understand that complex numbers can be represented on a plane called the complex plane. In this plane, the horizontal axis represents the real part, and the vertical axis represents the imaginary part. This representation enables a more comprehensive approach to solving equations that would otherwise not have solutions within the realm of real numbers.
The geometric interpretation of complex equations, such as the one in our exercise, aids in visualizing solutions. Each complex number corresponds to a unique point on this plane, making it easier to apply geometric reasoning.

Ellipse Geometry

Ellipses are defined as the set of points satisfying a particular geometrical condition. An ellipse can be thought of as stretched circles around its center, with two distinct axes: the major axis (the longest diameter) and the minor axis (the shortest diameter).
The standard definition of an ellipse involves the sum of the distances from any point on the ellipse to two fixed points, called foci, being constant. This condition is represented in the form: the sum of distances from a point on the ellipse to two foci is equal to the length of the major axis.
In our original problem, the expression \(2|z + 3i| = |z - i|\) aligns with this definition. Here, the complex number point \( z \) must satisfy this constant distance ratio, implying an ellipse as the solution. The length of the major axis in this context is calculated as \(\frac{16}{3}\). This aligns with the known property that the distances from the foci to any point on an ellipse must satisfy the equation.

Modulus of Complex Numbers

The modulus of a complex number \( z = a + bi \) is denoted as \(|z|\) and is a measure of its magnitude (distance from the origin on the complex plane). It is calculated using the formula \(|z| = \sqrt{a^2 + b^2}\), resembling the Pythagorean theorem.
In our problem, the modulus \(|z + a|\) implies measuring the distance between the point \( z \) and another complex number \( -a \). Similarly, \(|z - b|\) measures the distance from \( z \) to the complex number \( b \) in the plane.
By understanding the modulus, we can effectively resolve the exercise equation \(2|z + 3i| = |z - i|\), which describes a locus of points equidistantly scaled within a fixed ratio to two points. The modulus thus serves to enforce geometrical constraints, illustrating an ellipse's shape as formed by these distances.