Problem 128 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

The two circles \(x^{2}+y^{2}=a x\) and \(x^{2}+y^{2}=c^{2}(c>0)\) touch each other if (A) \(|a|=c\) (B) \(a=2 c\) (C) \(|a|=2 c\) (D) \(2|a|=c\)

Step-by-Step Solution

Answer

The circles touch if \(|a| = 2c\) (option C).

Step 1: Identify the Center and Radius of Circle 1

The equation of the first circle is \( x^2 + y^2 = ax \). This can be rewritten in standard circle form as \( (x - \frac{a}{2})^2 + y^2 = \frac{a^2}{4} \). The center is \((\frac{a}{2}, 0)\), and the radius is \(\frac{|a|}{2}\).

Step 2: Identify the Center and Radius of Circle 2

The equation of the second circle is \( x^2 + y^2 = c^2 \). This is in the standard form, centered at \((0, 0)\) with radius \(c\).

Step 3: Apply the Condition for Two Circles to Touch

Two circles touch each other if the distance between their centers is equal to the sum or difference of their radii. The distance between the centers \((\frac{a}{2}, 0)\) and \((0, 0)\) is \(\frac{|a|}{2}\).

Step 4: Write the Equation for Touching Circles

For the circles to touch externally, the distance between the centers must equal the sum of the radii: \(\frac{|a|}{2} + c = \frac{|a|}{2}\). Solving this equation gives \(|a| = 2c\).

Step 5: Choose the Correct Option

Based on our calculations, we determined that \(|a| = 2c\). Therefore, the correct answer is option (C).

Key Concepts

Touching Circles

When two circles are said to be touching each other, it means they meet at exactly one point. This can occur either externally or internally.

• Externally touching circles: The circles have one common point on their exteriors. Picture two circles just "kissing" each other from the outside. The center-to-center distance equals the sum of their radii.
• Internally touching circles: Here, one circle lies within another, again meeting at only one point. The center-to-center distance is the difference between their radii.

Understanding when two circles touch is essential in problems involving circle placement and motion, allowing us to solve equations based on their geometric relations.

Circle Radius

The radius of a circle is the constant distance from the center to any point along the circle's boundary. In mathematical terms, it is half the diameter, sometimes represented by the constant \\[r = \frac{d}{2}\].

Understanding how to determine the radius from a circle's equation is crucial:

• For the circle equation in the standard form \\[(x - h)^2 + (y - k)^2 = r^2\], the radius is \\[r\].
• From the equation \\[x^2 + y^2 = ax\], rewrite it into \\[(x - \frac{a}{2})^2 + y^2 = (\frac{a}{2})^2\] to identify the radius as \\[(\frac{|a|}{2})\].
• For the circle equation \\[x^2 + y^2 = c^2\], the radius \\[r\] is simply \\[c\].

Grasping these principles facilitates solving various geometric problems involving circles.

Circle Equation

The equation of a circle provides valuable information about its geometry, helping to describe its placement and dimension. A standard circle equation looks like \\[(x-h)^2 + (y-k)^2 = r^2\], where:

• \(h, k\) are the coordinates of the circle's center.
• \(r\) is the radius.

The equation reveals both the positioning and size of the circle. By translating any given circle equation into this standard form, you can easily find both the center and radius, aiding in further geometric applications.

In our example, the first given circle \\[x^2 + y^2 = ax\] was transformed into \\[(x - \frac{a}{2})^2 + y^2 = (\frac{a}{2})^2\], revealing a center at \\[(\frac{a}{2}, 0)\] and radius \\[\frac{|a|}{2}\]. The second circle maintained its standard form \\[x^2 + y^2 = c^2\], with a center at the origin \\[(0, 0)\] and radius \\[c\].

Converting equations into this format is a fundamental skill in geometry, enabling deeper understanding and analysis of circle problems.