Problem 8 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

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

Step-by-Step Solution

Answer

The answer is (a) \(y^2 = x^2 + 2xy \frac{dy}{dx}\).

Step 1: Understand the Circle's Equation

The general equation for a circle with its center at \((x_0, 0)\) and passing through the origin is given by \((x - x_0)^2 + y^2 = x_0^2\). This simplifies to \(x^2 - 2x_0x + y^2 = 0\). Our task is to find the differential equation that represents all such circles.

Step 2: Differentiate the General Circle Equation

Differentiate the equation \(x^2 - 2x_0x + y^2 = 0\) with respect to \(x\). This yields: \[2x - 2x_0 + 2y \frac{dy}{dx} = 0\]Simplifying, we get: \[x - x_0 + y \frac{dy}{dx} = 0\]

Step 3: Eliminate the Circle Parameter

From the equation \(x - x_0 + y \frac{dy}{dx} = 0\), solve for \(x_0\):\[x_0 = x + y \frac{dy}{dx}\]Since the equation must hold for any value of \(x_0\), eliminate \(x_0\) using its expression in terms of \(x\), \(y\), and \(\frac{dy}{dx}\).

Step 4: Simplify and Rearrange

Substitute \(x_0 = x + y \frac{dy}{dx}\) back into the equation \((x - x_0)^2 + y^2 = x_0^2\) to get:\[x^2 - 2x(x + y \frac{dy}{dx}) + y^2 = 0\]Simplify to form: \[x^2 - 2x^2 - 2xy \frac{dy}{dx} + y^2 = 0\], which becomes \[y^2 = x^2 + 2xy \frac{dy}{dx}\].

Key Concepts

Circles

Circles are shapes where all points are equidistant from a center point. This distance is known as the radius. A circle's equation expresses the relationship between the coordinates of points on the circle and its center.For any general circle centered at \((x_0, 0)\) on the x-axis and passing through the origin, we use the equation \((x - x_0)^2 + y^2 = x_0^2\). This formula is handy because it helps to visually understand how the circle interacts with the coordinate plane.A key feature of circles relevant to our problem is that they have their centers on the x-axis. This specific configuration leads to equations that can be simplified to represent many such circles. In the context of calculus and differential equations, identifying these equations allows us to tackle more complex mathematical problems.

Calculus

Calculus is a branch of mathematics focusing on change and motion. It contains tools for differentiation and integration that are immensely helpful in understanding and describing various real-world scenarios, such as motion and growth.Differentiation is one such tool. It's used to find slopes of curves or to understand how quantities change with respect to each other. In this context, we differentiate the circle’s equation to obtain a derivative, \(\frac{dy}{dx}\), which describes how the circle changes with respect to x.By differentiating the equation \(x^2 - 2x_0x + y^2 = 0\), we gain insights into the dynamic nature of the circle’s radius and how it interacts with changing coordinates. The result, \(2x - 2x_0 + 2y \frac{dy}{dx} = 0\), incorporates the concept of the rate of change directly into our understanding of the circle's properties.

Coordinate Geometry

Coordinate Geometry, or analytic geometry, combines algebra and geometry using a coordinate system. It's a detailed approach to position shapes on a plane. Each point is marked by its coordinates (x, y), which represent its location relative to two perpendicular lines called axes. In our exercise, coordinate geometry helps us localize circles specifically. By setting their centers on the x-axis, we simplify their general equation. Coordinate geometry then makes it possible to transform this geometric setup into algebraic expressions that can be analyzed using calculus. In solving for all circles through the origin and with centers on the x-axis, we express the circles’ equations in a simplified form which can then be differentiated. This demonstrates how coordinate geometry helps to distill complex shapes into manageable mathematical expressions. Ultimately, it enhances our ability to find and describe different possible configurations of these circles through equations.