Problem 107 - A Complete Resource Book in Mathematics for JEE Main 2017

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 centers lie on a hyperbola.

Step 1: Identify and Re-center the Given Circle

The given equation of the circle is \(x^2 + y^2 - 8x - 8y - 4 = 0\). To find the center and radius, complete the square for both \(x\) and \(y\). \((x-4)^2 + (y-4)^2 = 36\) shows the center at \((4,4)\) and radius \(6\).

Step 2: Consider Circle Tangent to x-axis

A circle that is tangent to the \(x\)-axis will have its center at \((h, k)\) such that \(k = r'\), where \(r'\) is its radius, to barely touch the \(x\)-axis from above.

Step 3: External Tangency Condition

The circle is externally tangent to the given circle, which means the distance between the centers equals the sum of the two radii. Therefore, \(\sqrt{(h - 4)^2 + (k - 4)^2} = 6 + r'\).

Step 4: Solve the Constraints

From Step 2, since the circle touches the \(x\)-axis at \(k\), we have \(k = r'\). Substitute \(r'\) with \(k\) into the equation in Step 3, giving \(\sqrt{(h-4)^2 + (k-4)^2} = 6 + k\). Simplifying, we get \((h-4)^2 + (k-4)^2 = (6+k)^2\).

Step 5: Determine the Locus of Centers

The equation from Step 4 simplifies to \(h^2 + k^2 - 8h - 8k + 16 = k^2 + 12k + 36\). Further simplification yields \(h^2 - 8h - 8k = 12k + 20\), which is equivalent to \((h - 4)^2 - (k + 2)^2 = -4\). This fits the form of a hyperbola.

Key Concepts

Circle Tangency

The concept of circle tangency is fundamental in understanding how two circles can interact with each other. Tangency refers to the point or line where two shapes meet without intersecting each other. When discussing circles, there are two main types of tangency to consider: **external tangency** and **internal tangency**.
External tangency occurs when two circles touch each other from the outside. This means the circles share a single point on their boundaries, but do not overlap. The distance between the centers of the circles is equal to the sum of their radii.
When comparing this with the exercise, we consider circles that must be tangential to both another circle and the x-axis. **The condition for external tangency** is critical in determining where the centers of these circles lie. 🌐 This leads us to consider how each circle's position is precisely determined by this tangency condition. Students should focus on these specific requirements to grasp the concept and solve similar problems.
Some key points:

• The point of tangency is where the circle meets the line (or another circle).
• For external tangency, the sum of radii of both circles equals the distance between the centers.
• Circle tangency is a useful way to identify the locus of potential center points for other circles.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is the branch of geometry where we use a coordinate system to explore geometric figures. It allows us to calculate distances, angles, and conic sections by using algebraic equations. In this problem, we're employing coordinate geometry to analyze and solve the conditions for circle tangency.
The given circle’s equation, \(x^2 + y^2 - 8x - 8y - 4 = 0\) can be transformed through completing the square, revealing its center and radius. Coordinate geometry provides us the method to calculate these details accurately.

• Start by recentering the circle using the process of completing the square.
• For circles touching axes, adjust coordinates to reflect this constraint – use 'k' for representation as needed.
• Understand the symmetry and transformations that can occur within this coordinate plane.

Using coordinate geometry, we seamlessly move from graphical representation to an algebraic format, providing a powerful toolkit for analyzing complex geometric relationships.

Locus of Points

The term 'locus of points' refers to a set of points satisfying a particular condition or a group of related conditions. In our exercise, the locus is comprised of the possible centers \((h, k)\) of the circles that meet the tangency criteria: external tangency to the given circle, and tangency to the x-axis.
Consider this locus as a path traced by points \((h, k)\) that when connected, form a specific shape based on the geometrical constraints provided. From our solution, we find that the locus of the centers fitting all conditions is a hyperbola.
Key aspects of loci include:

• Loci define possible positions— especially valuable for dynamic conditions like tangency.
• Mathematically expressed as functions or equations like the hyperbolic equation deduced here.
• Understanding how changing one aspect of a shape (radius, center position) affects the entire locus is crucial.

The simplification of the equation \((h - 4)^2 - (k + 2)^2 = -4\) confirms whether these centers form known geometric shapes such as hyperbolas or other conics, enabling students to visualize and solve complex geometry problems.