Problem 107 - A Complete Resource Book in Mathematics for JEE Main

Question

The centres of those circles which touch the circle, \(x^{2}\) \(+y^{2}-8 x-8 y-4=0\), externally and also touch the \(x\)-axis, lie on (A) A parabola (B) A circle (C) An ellipse which is not a circle (D) A hyperbola

Step-by-Step Solution

Answer

The locus is a hyperbola (option D).

Step 1: Recognize the Circle Equation in Standard Form

First, identify the center and radius of the given circle by rewriting the equation to complete the square. The original circle equation is \( x^2 + y^2 - 8x - 8y - 4 = 0 \).

Step 2: Complete the Square

Rewrite the equation in the form \((x-h)^2 + (y-k)^2 = r^2\). Start by completing the square. Rearrange terms: \( x^2 - 8x + y^2 - 8y = 4 \) Complete the square for \(x\): \( (x^2 - 8x) = (x-4)^2 - 16 \)Complete the square for \(y\): \( (y^2 - 8y) = (y-4)^2 - 16 \)Substitute these back into the equation:\( (x-4)^2 - 16 + (y-4)^2 - 16 = 4 \)\( (x-4)^2 + (y-4)^2 = 36 \)

Step 3: Identify Center and Radius

From \((x-4)^2 + (y-4)^2 = 36\), we can see that the center of the circle is at \((4, 4)\) and the radius is \(\sqrt{36} = 6\).

Step 4: Determine Conditions for the Locus of Centers

For another circle to touch this circle externally and the \(x\)-axis, its center \((h, k)\) must be at a distance \(r + 6\) from \((4, 4)\), and the circle must have radius \(k\) to touch the \(x\)-axis.

Step 5: Set Up the Locus Equation

Apply the distance formula between the centers and the condition to touch the \(x\)-axis. The equation for the distance between centers is:\((h-4)^2 + (k-4)^2 = (r+6)^2\).Since the circle touches the \(x\)-axis, radius \(r = k\).Substitute: \((h-4)^2 + (k-4)^2 = (k+6)^2 \).

Step 6: Simplify Locus Equation

Expand the equation:\( (h-4)^2 + (k-4)^2 = k^2 + 12k + 36 \)Re-collect terms:\( h^2 - 8h + 16 + k^2 - 8k + 16 = k^2 + 12k + 36 \)Simplify to get:\( h^2 - 8h = 20k + 4 \)

Step 7: Analyze the Curve Equation

Reorganize: \( h^2 - 8h - 20k = 4 \).Compare with the standard form of a conic section. The absence of \(k^2\) and the presence of \(h\) and \(k\) terms with different coefficients indicate a hyperbola.

Key Concepts

Conic Sections

Conic sections are fascinating curves formed by the intersection of a plane with a double-napped cone. They include shapes like circles, ellipses, parabolas, and hyperbolas. Each of these shapes has distinct geometric properties:

• **Circle:** All points are equidistant from a central point.
• **Ellipse:** The sum of distances from two fixed points, called foci, remains constant for every point on the curve.
• **Parabola:** Contains all points equidistant from a point and a line. The focus lies at the point and the directrix is the line.
• **Hyperbola:** The difference of distances from two foci remains constant, forming two open curves.

Circles are actually special cases of ellipses where the foci are at the same location, resulting in a perfect round shape. When studying conic sections, finding relationships between these curves can help predict and explain geometric and algebraic properties.

Circle Equations

The equation of a circle in its standard form is: \[(x-h)^2 + (y-k)^2 = r^2\]Here,

• \((h, k)\) is the center of the circle,
• \(r\) is the radius.

The exercise presented involves converting an expanded form into this standard form through a process called completing the square. This process helps us quickly identify the center and radius of the circle. In this particular problem, the circle was defined by the equation: \[x^2 + y^2 - 8x - 8y - 4 = 0\]By completing the square, the equation transforms into \[(x-4)^2 + (y-4)^2 = 36\]indicating that the circle's center is \((4,4)\) and its radius is \(6\).Understanding how to manipulate and read circle equations is essential for solving problems involving circles and other conic sections.

Locus of Centers

The locus of centers refers to a collection of points satisfying certain conditions, often describing a path or shape in geometry. In problems like this exercise, we are dealing with finding a path for the centers of circles that satisfy given conditions. Here, we have two conditions:

• The centers must be a certain distance from a given circle's center.
• The circles must also touch the x-axis.

With these dual conditions, we derive an equation using geometry and algebra. For our specific problem, the condition for the external touch with circle \((x-4)^2 + (y-4)^2 = 36\) is that the center must be a distance \(r + 6\) from the circle's center, \((4, 4)\). Also, for the circle to touch the x-axis, the radius must equal the y-coordinate of the center; hence the center is \((h, k)\).These conditions translated into the equation: \[(h-4)^2 + (k-4)^2 = (k+6)^2\].Solving this leads us to find the path the centers can lie on, revealing insights about the nature of the conic section.

Parabola, Hyperbola, Ellipse

In conic sections, ellipses, parabolas, and hyperbolas each have unique structures. Understanding when and why a locus forms these shapes is crucial. The original problem, through analysis of conditions enabled by the circles' paths, results in identifying a hyperbola. Here's why:

• A **parabola** is not identified because it involves only one squared term typically.
• An **ellipse**, similar to a circle, involves balancing squared terms equally, which was not the case here.
• The absence of a squared term in one variable and unequal coefficients marks a **hyperbola**.

This problem's equation \[h^2 - 8h - 20k = 4\]lacks a squared term for both variables, a distinctive characteristic of a hyperbola. These calculations derive from conic sections' standard forms and properties. When circles, interacting with axes and each other, create paths of centers, the confinement to specific conic shapes showcases the inherent geometric harmony. Recognizing a hyperbola enriches the understanding of how geometric constraints guide transformations and shift conceptual perspectives in solving mathematical problems.