Problem 188 - A Complete Resource Book in Mathematics for JEE Main

Question

Assertion: If \(|z-1|+|z+3| \leq 8\), then the range of values of \(|z-4|\) is \([1,9]\) Reason: \(|z-1|+|z+3| \leq 8 \Rightarrow z\) lies inside or on the ellipse whose foci are \((1,0)\) and \((-3,0)\) and vertices are \((-5,0)\) and \((3,0)\).

Step-by-Step Solution

Answer

Assertion is correct; reason correctly explains the assertion.

Step 1: Understand the Assertion

The assertion states that if \(|z-1| + |z+3| \leq 8\), then the range of \(|z-4|\) is \([1,9]\). This can be represented geometrically as the set of points \(z\) satisfying \(|z-1| + |z+3| \leq 8\), which describes an ellipse with foci at \((1, 0)\) and \((-3, 0)\).

Step 2: Visualize the Geometry

The condition \(|z-1| + |z+3| \leq 8\) describes an ellipse. Its foci are at \((1,0)\) and \((-3,0)\), and the sum of distances from any point on this ellipse to the foci is constant (8 in this case). Hence, the ellipse lies between its end vertices \((-5,0)\) and \((3,0)\).

Step 3: Determine Range of Values for \(|z-4|\)

To find \(|z-4|\), we locate the point \((4,0)\) relative to the ellipse. The maximum distance from \(z\) on the ellipse to \((4,0)\) occurs when \(z\) is at \((-5,0)\), and the minimum occurs when \(z\) is at \((3,0)\). Therefore, calculate these distances: \(|-5 - 4| = 9\) and \(|3 - 4| = 1\).

Step 4: Conclusion for Assertion

Since the distances have been calculated to be between 1 and 9, the range of \(|z-4|\) for points \(z\) on the ellipse or within it is indeed \([1, 9]\). Thus, the assertion is correct.

Step 5: Evaluate the Reason

The reason correctly identifies the condition \(|z-1| + |z+3| \leq 8\) as describing an ellipse with given foci and vertices, supporting the assertion. Therefore, both the assertion and reason are true, and the reason correctly explains the assertion.

Key Concepts

Geometry of Complex Numbers

Complex numbers are not just abstract quantities but have geometric significance on the complex plane. Any complex number can be represented as a point or a vector in this plane, which consists of a real part and an imaginary part. This gives a straightforward method to visualize complex numbers geometrically.
For example, consider a complex number represented as \( z = x + yi \), where \( x \) is the real component, and \( y \) is the imaginary component. This can be plotted as a point \( (x, y) \) in the complex plane, essentially a 2D coordinate system.
A critical application of the geometry of complex numbers is in finding distances. By calculating the modulus, or absolute value, of a complex number, we find the distance of the number from the origin: \(|z| = \sqrt{x^2 + y^2}\). This geometric interpretation aids immensely in solving problems involving loci, such as circles and ellipses.
When questions involve conditions like \(|z-a| + |z-b| \leq c\), they lead to the formation of curves like ellipses when considered geometrically, as alluded to in our exercise.

Ellipses in Complex Plane

Ellipses on the complex plane can be constructed using the concept of distances to foci. An ellipse is characterized by two fixed points called foci, where the sum of the distances from any point on the ellipse to these foci is constant.
In our exercise, the condition \(|z-1| + |z+3| \leq 8\) helps in visualizing an ellipse. The foci of this ellipse are at the points \((1, 0)\) and \((-3, 0)\) in the complex plane.
The effective locus created by this condition has its vertices at \((-5, 0)\) and \((3, 0)\), making this a horizontal ellipse centered along the real axis. The total span along the x-axis is determined by the maximum stretch of the ellipse, which occurs within these limits.
Understanding ellipses in this way becomes immensely helpful in relating algebraic conditions in complex numbers to their geometric counterparts, helping solve such problems intuitively and efficiently.

Distance Formula in Complex Plane

When dealing with complex numbers, the distance formula is an essential tool. It helps determine how far points (or complex numbers) are from each other in the complex plane—crucial for analyzing geometric problems.
Given two complex numbers, \(z_1 = x_1 + y_1i\) and \(z_2 = x_2 + y_2i\), their distance can be calculated with the formula \(|z_1 - z_2| = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\). This formula mirrors the Euclidean distance found in basic geometry.
In our context of the problem, the distance \(|z-4|\) essentially encapsulates how far a complex number \(z\) is from the point \((4,0)\) on the real axis. By assessing this distance for points on the ellipse described by \(|z-1| + |z+3| \leq 8\), we discover the range of potential values, delineated as \[1, 9\].
Hence, leveraging the distance formula equips us to navigate through the complex plane seamlessly and translate intricate algebraic expressions into clear geometric interpretations.