Problem 74 - A Complete Resource Book in Mathematics for JEE Main

Question

The equation of a circle of radius 2 touching the circles \(x^{2}+y^{2}-4|x|=0\) is (A) \(x^{2}+y^{2}+2 \sqrt{3} y+2=0\) (B) \(x^{2}+y^{2}+4 \sqrt{3} y+8=0\) (C) \(x^{2}+y^{2}-4 \sqrt{3} y+8=0\) (D) none of these

Step-by-Step Solution

Answer

The correct answer is (C).

Step 1: Identifying the Given Circle

The given circle is described by the equation \(x^2 + y^2 - 4|x| = 0\). We need to find the centers and radii of the circle it represents. By rewriting the equation for positive \(x\), we have \(x^2 + y^2 = 4x\), implying center at \((2, 0)\) with radius 2. For negative \(x\), rewrite as \(x^2 + y^2 = -4x\) implying center at \((-2, 0)\) and radius 2.

Step 2: Understanding the Ring Configuration

The equation represents two circles, one with center \((2, 0)\) and another with \((-2, 0)\), both having radius 2. They touch each other at the origin, forming a 'ring' configuration. Our task is to find a circle with radius 2 that touches either of these circles.

Step 3: Analyzing Touch Conditions

To touch the circle with center \((2, 0)\), a new circle (radius 2) can have its center at \((2, y_0)\). The distance between centers must equal the sum of radii: \(4 = 2 + 2\); thus possible center scenarios include \((2, \pm 2\sqrt{3})\), above and below the line \(x = 2\).

Step 4: Equation of the Candidate Circles

Using the possible centers \((2, \pm 2\sqrt{3})\), we write equations of candidate circles with radius 2:1. For the circle above: \((x - 2)^2 + (y - 2\sqrt{3})^2 = 4\).2. For the circle below: \((x - 2)^2 + (y + 2\sqrt{3})^2 = 4\).

Step 5: Expanding and Simplifying Equations

Expanding the circle equations:1. \((x - 2)^2 + (y - 2\sqrt{3})^2 = 4\) simplifies to \(x^2 + y^2 - 4x - 4\sqrt{3}y + 8 = 0\).2. \((x - 2)^2 + (y + 2\sqrt{3})^2 = 4\) simplifies to \(x^2 + y^2 - 4x + 4\sqrt{3}y + 8 = 0\).Comparing these, the only matching option given is \(x^2 + y^2 - 4\sqrt{3}y + 8 = 0\).

Step 6: Selecting the Correct Option

The correct option based on simplification is \(x^2 + y^2 - 4\sqrt{3}y + 8 = 0\) which corresponds to option (C) in the given list.

Key Concepts

Equation of a Circle

The equation of a circle is a mathematical expression that represents all the points equidistant from a central point, known as the circle's center. A commonly used form to express this equation is \[ (x - h)^2 + (y - k)^2 = r^2 \]where:

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

This formula helps define the circle's location and size.
When we expand or manipulate this equation, we may arrive at various standard forms while expressing the same circle properties. For example, when a circle's equation is given in expanded form, like \(x^2 + y^2 + Dx + Ey + F = 0\), we can often complete the square to revert to the standard form.
Understanding the equation of a circle enables us to derive other important properties, such as distance from a point to the circle or the interaction with other geometric figures, like other circles.

Distance Between Centers

The distance between the centers of two circles is an important concept when exploring their geometric relationships. Calculating this distance helps understand how circles are positioned relative to each other.
Given two circles with centers at \((x_1, y_1)\) and \((x_2, y_2)\), we calculate the distance using the Euclidean distance formula:\[\text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]
This concept is particularly useful in problems where circles intersect, touch, or are entirely separate. It helps in determining

• Whether two circles are touching externally.
• If they overlap, based on radii comparisons.
• In what configuration they lie.

For example, if the distance equals the sum of the radii, the circles touch externally. Understanding this distance is fundamental in solving related geometric puzzles.

Radius of a Circle

The radius of a circle is the constant distance from the center of the circle to any point on its edge. It is a crucial measure that defines a circle's size and is denoted by \(r\).
The radius plays a pivotal role in determining a circle's equation and other properties, such as its circumference or area. For instance, if you know the radius, you can easily find the circumference using the formula:\[\text{Circumference} = 2\pi r\]Similarly, the area is given by:\[\text{Area} = \pi r^2\]
When analyzing interactions with other circles, the radius becomes key to determining how they can touch or intersect. If two circles have specific radii, the relationship between these radii and the distance between centers can tell us much about how they spatially relate to one another.