Problem 71 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

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

(c) \(|a|=2c\)

Step 1: Identify Circle Equations and Interpret

The two given circles are 1. Circle 1: \(x^2 + y^2 = ax\) with center \( (\frac{a}{2}, 0) \) and radius \( \frac{|a|}{2} \).2. Circle 2: \(x^2 + y^2 = c^2\) with center at \( (0, 0) \) and radius \( c \).We need to determine when these circles touch each other, meaning their distance equals the sum or difference of their radii.

Step 2: Calculate Distance Between Centers

The distance between the centers of the two circles is calculated using the formula for distance between two points: \[\text{Distance} = \sqrt{\left( \frac{a}{2} - 0 \right)^2 + (0 - 0)^2 } = \frac{|a|}{2}\]

Step 3: Consider Conditions for Touching Circles

Circles touch if the distance between their centers is equal to the sum or difference of their radii.Thus, - Touching from outside: \( \frac{|a|}{2} + c = c \)- Touching from inside: \( \left| \frac{|a|}{2} - c \right| = 0 \)

Step 4: Determine Conditions

Since the circles touch each other from inside,\(\left| \frac{|a|}{2} - c \right| = 0\)\(\frac{|a|}{2} = c\)Solving gives \(|a|=2c\). Thus, the correct condition is (c).

Key Concepts

Circle Equations

Understanding the equations of circles is foundational in solving geometry problems involving touching circles. In general, the standard equation of a circle in a coordinate plane is given as \(x^2 + y^2 = r^2\), where \(r\) is the radius of the circle. For the two circles presented, the equations are slightly modified:

• Circle 1: \(x^2 + y^2 = ax\) is centered at \((\frac{a}{2}, 0)\) with radius \(\frac{|a|}{2}\).
• Circle 2: \(x^2 + y^2 = c^2\) is centered at the origin \((0,0)\) with radius \(c\).

Identifying the center and radius from these equations is crucial, as it sets the stage for the subsequent calculations needed to determine conditions for the circles touching each other. The manipulations of these equations involve leveraging algebraic techniques to derive and interpret the distances and relationships between these circles.

Distance Between Centers

The geometry of circles often requires calculating the distance between the centers. This sets the baseline for comparison with the sum or difference of radii to determine if circles touch. In our problem, we use the distance formula between two points \((x_1, y_1)\) and \((x_2, y_2)\), which is given by \(\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}\).
Applying this formula here, the center of Circle 1 is \((\frac{a}{2}, 0)\), and the center of Circle 2 is \((0,0)\). Thus, the distance \(d\) between them is calculated as:\[ d = \sqrt{\left(\frac{a}{2} - 0\right)^2 + (0 - 0)^2 } = \frac{|a|}{2}\]
This calculation is a simple yet vital step, as it allows us to establish the critical relationship between circle positioning needed for touching conditions.

Geometry Problems

Geometry problems involving circles often require understanding several key concepts like radii, distance, and touching points. When two circles touch each other, they can do so in one of two ways:

• They can touch externally, where the distance between their centers is equal to the sum of their radii.
• They can touch internally, where the distance between their centers is equal to the absolute difference of their radii.

In the context of our problem, we examine when these circles will touch internally. By equating the distance between centers to the difference in radii, we ensure that solutions satisfy the condition of touching circles. Understanding these geometry concepts deepens problem-solving skills, particularly for intricate configurations like internally or externally tangent circles.

Circle Properties

Circles present distinct properties that govern their interaction with each other. For circles to touch — whether externally or internally — certain mathematical properties come into play:

• Radius: Defines the size of the circle, impacting likelihood and nature of touching.
• Center: The fixed point from which the radius is measured, crucial for determining relative positions.

When solving for when two circles touch, recognizing the relation between their radii and the distance between their centers is essential. For example, given \(\frac{|a|}{2} = c\), the properties and configuration satisfy internal touching conditions in the problem. Having clarity on circle properties aids in visualizing and solving varied geometry problems, ensuring robust comprehension and application to real-world scenarios.