Problem 85 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

Intercept on the line \(y=x\) by the circle \(x^{2}+y^{2}-2 x=0\) is \(A B\). Equation of the circle on \(A B\) as a diameter is (a) \(x^{2}+y^{2}+x-y=0\) (b) \(x^{2}+y^{2}-x+y=0\) (c) \(x^{2}+y^{2}+x+y=0\) (d) \(x^{2}+y^{2}-x-y=0\)

Step-by-Step Solution

Answer

The answer is option (d): \(x^2 + y^2 - x - y = 0\).

Step 1: Find Points of Intersection

Substitute the line equation \( y = x \) into the circle equation \( x^2 + y^2 - 2x = 0 \). This gives \( x^2 + x^2 - 2x = 0 \) or \( 2x^2 - 2x = 0 \). Simplify to \( 2x(x - 1) = 0 \) leading to \( x = 0 \) or \( x = 1 \). Since \( y = x \), the points of intersection are \( (0, 0) \) and \( (1, 1) \).

Step 2: Determine the Equation of the Circle with Diameter AB

The endpoints of diameter \( AB \) are \( (0, 0) \) and \( (1, 1) \). The midpoint \((h, k)\) of \( AB \) is \( \left( \frac{0+1}{2}, \frac{0+1}{2} \right) = \left( \frac{1}{2}, \frac{1}{2} \right) \). The radius squared \( r^2 \) is \( \left( \frac{1}{2} - 0 \right)^2 + \left( \frac{1}{2} - 0 \right)^2 = \frac{1}{2} \). Thus, the equation of the circle is \( (x - \frac{1}{2})^2 + (y - \frac{1}{2})^2 = \frac{1}{2} \).

Step 3: Simplify the Circle Equation to Match Given Options

Expand the circle equation: \( (x - \frac{1}{2})^2 + (y - \frac{1}{2})^2 = \frac{1}{2} \) becomes \( x^2 - x + \frac{1}{4} + y^2 - y + \frac{1}{4} = \frac{1}{2} \). Combine terms to get \( x^2 + y^2 - x - y + \frac{1}{2} = \frac{1}{2} \). Cancel \( \frac{1}{2} \) on both sides to simplify to \( x^2 + y^2 - x - y = 0 \).

Step 4: Select the Correct Option

Compare the simplified circle equation \( x^2 + y^2 - x - y = 0 \) with the provided options. The matching option is (d) \( x^2 + y^2 - x - y = 0 \).

Key Concepts

Circle Equations

The equation of a circle is a key element in coordinate geometry. When we describe a circle in the xy-plane, we typically use the standard form equation \[(x - h)^2 + (y - k)^2 = r^2\]where

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

In this exercise, we used a specific example of a circle equation, \[x^2 + y^2 - 2x = 0\]which represents a circle centered at \((1, 0)\) with radius \(1\). The circle's equation can be rewritten in standard form by completing the square, which is a useful method for converting equations into a recognizable circle form. Understanding how to manipulate and interpret these equations helps in solving various geometric and mathematical problems. Circle equations play an essential role in determining intersections, tangents, and other properties connected to circles in coordinate geometry.

Line Intercepts

Line intercepts refer to the points at which a line crosses the x-axis and y-axis. In this context, we're discussing the intercept of a line with a circle. For the given problem, the line given by the equation \[y = x\] intersects the circle \[x^2 + y^2 - 2x = 0\]. Performing substitution to find the points of intersection, we set \[y = x\] into the circle's equation, yielding \[2x^2 - 2x = 0\]. Solving this gives \[x(2x - 2) = 0\], leading to two intercept points, \((0, 0)\) and \((1, 1)\). These points are significant as they form the diameter endpoints for another circle. This process of finding intercepts is essential for interpreting relationships between various geometric entities in coordinate geometry.

Coordinate Geometry

Coordinate geometry, or analytic geometry, combines algebra and geometry to solve problems involving points, lines, and shapes on a coordinate plane. It allows the determination of geometric properties through algebraic equations. For example, given points \((0, 0)\) and \((1, 1)\), you can determine important information like the midpoint \((\frac{1}{2}, \frac{1}{2})\), which is used as the center of a circle having these points as its diameter. The concept that the midpoint of a line segment serves as a pivotal calculation in forming and determining the properties of geometric shapes is a foundational aspect of coordinate geometry. Additionally, expanding upon algebraic manipulation and simplification, you gain insights into geometric relationships and can solve complex problems, such as finding a specific equation for a circle based on given conditions. Coordinate geometry is a versatile tool in mathematics, connecting algebraic expressions with geometric visualization and problem-solving.