Problem 108 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

The differential equation of all circles passing through the origin and having their centres on the \(x\)-axis is (A) \(x^{2}=y^{2}+x y \frac{d y}{d x}\) (B) \(x^{2}=y^{2}+3 x y \frac{d y}{d x}\) (C) \(y^{2}=x^{2}+2 x y \frac{d y}{d x}\) (D) \(y^{2}=x^{2}-2 x y \frac{d y}{d x}\)

Step-by-Step Solution

Answer

The correct differential equation is option (D): \(y^2 = x^2 - 2xy \frac{dy}{dx}\).

Step 1: Understanding the Problem

A circle with its center on the x-axis can be described using the equation \((x-a)^2 + y^2 = a^2\), where \(a\) is the x-coordinate of the center. Since it passes through the origin, substituting \((x, y) = (0, 0)\) in the circle equation gives \(a^2 + 0^2 = a^2\), confirming that it passes through the origin.

Step 2: Finding the General Equation

From the equation of a circle \((x-a)^2 + y^2 = a^2\), expand it to \(x^2 - 2ax + a^2 + y^2 = a^2\). Simplifying, this yields \(x^2 + y^2 = 2ax\). This is the general equation for circles passing through the origin with centers on the x-axis.

Step 3: Differentiation Step

Differentiate both sides of the equation \(x^2 + y^2 = 2ax\) with respect to \(x\). The left-hand side becomes \(2x + 2y\frac{dy}{dx}\) and the right-hand side becomes \(2a\). Thus, the derivative form becomes \(2x + 2y\frac{dy}{dx} = 2a\).

Step 4: Eliminate the Parameter

Since \(2a = x^2 + y^2\), substitute this expression back into the differentiated form: \(2x + 2y\frac{dy}{dx} = x^2 + y^2\). Rearrange to obtain: \(x^2 + y^2 = 2xy\frac{dy}{dx}\).

Step 5: Simplify to Desired Form

Re-arranging the result \(x^2 + y^2 = 2xy\frac{dy}{dx}\) gives us \(y^2 = x^2 - 2xy\frac{dy}{dx}\). This matches the given option \(D\).

Key Concepts

Circle Equation

A circle's equation is a mathematical representation that defines all the points equidistant from a central point in a plane. It is typically given by the standard form \( (x - h)^2 + (y - k)^2 = r^2 \), where

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

In the context of our exercise, we are dealing with circles where the center is specifically located on the x-axis. This can be described by the modified equation \((x-a)^2 + y^2 = a^2\), where \(a\) is the x-coordinate of the center. When a circle passes through the origin, substituting \((x, y) = (0, 0)\) gives a special condition that holds true for all such circles.This equation is important because circles exhibit symmetry and this equation helps retain that symmetrical property essential for geometric studies.

Differentiation

Differentiation in calculus is a fundamental operation that provides insights into how functions change. In simpler words, it tells us the rate at which one quantity changes with respect to another. In our exercise, the equation of the circle \(x^2 + y^2 = 2ax\) is differentiated with respect to the variable \(x\). This helps us find how the circle's circumference changes as the x-coordinate changes. This derivative allows the relationship between x and y to be further explored. The differentiation leads to the expression \(2x + 2y\frac{dy}{dx} = 2a\). Performing differentiation on such geometric equations is crucial because it helps identify important characteristics like slope and concavity, which are vital for understanding the behavior of curves and surfaces.

Geometric Loci

Geometric loci refer to the set of points that satisfy certain geometric conditions. In simple terms, it's a collection of points that share a specific property or rule. For circles, the locus is the set of all points that are a fixed distance from a center point. In the context of this exercise, we are interested in the loci of all circles passing through the origin with centers on the x-axis. These points form a pattern or a path which in the real world allows one to understand the positioning and shape of geometric figures based on specific conditions. The idea of mapping loci is fundamental in both mathematical theory and practical applications, as it forms the basis for much of the study in fields such as physics and engineering, where relationships between variables are explored through spatial and geometric perspectives.