Problem 11 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

The differential equation for the family of circle \(x^{2}+y^{2}-2 a y=0\), where a is an arbitrary constant is (a) \(\left(x^{2}+y^{2}\right) y^{\prime}=2 x y\) (b) \(2\left(x^{2}+y^{2}\right) y^{\prime}=x y\) (c) \(\left(x^{2}-y^{2}\right) y^{\prime}=2 x y\) (d) \(2\left(x^{2}-y^{2}\right) y^{\prime}=x y\)

Step-by-Step Solution

Answer

Option (a) \((x^2 + y^2)y' = 2xy\) is the differential equation for the given family of circles.

Step 1: Identify the Original Equation

We start with the given equation of the circle, which is \(x^2 + y^2 - 2ay = 0\). This equation represents a family of circles centered on the y-axis with radius dependent on \(a\).

Step 2: Differentiate the Given Equation Implicitly

Differentiate the equation \(x^2 + y^2 - 2ay = 0\) with respect to \(x\) implicitly. We do this because we want to find an expression involving \(y'\), the derivative of \(y\) with respect to \(x\). Differentiating, we get: \(2x + 2y \cdot y' - 2a \cdot y' = 0\).

Step 3: Solve for the Derivative y'

From the differentiated equation, we solve for \(y'\):\(2x + 2y y' - 2a y' = 0\).Rearrange terms to isolate \(y'\):\(2y y' - 2a y' = -2x\).Factor out \(y'\):\(y'(2y - 2a) = -2x\).Thus, \(y' = \frac{-2x}{2y - 2a}\).

Step 4: Express y' in Terms of x and y Only

Since we initially had \(x^2 + y^2 = 2ay\), substitute \(a = \frac{x^2 + y^2}{2y}\) into the expression for \(y'\):\(y' = \frac{-2x}{2y - 2\left(\frac{x^2 + y^2}{2y}\right)}\).Simplifying the denominator:\(2y - 2\left(\frac{x^2 + y^2}{2y}\right) = 2y - \left(\frac{x^2 + y^2}{y}\right) = \frac{2y^2 - x^2 - y^2}{y}\).Thus, \(y' = \frac{-2xy}{-x^2 - y^2 + 2y^2} = \frac{2xy}{x^2 + y^2}\).

Step 5: Formulate the Differential Equation

Rewriting the expression for \(y'\) in a form that matches the provided options, we find:\((x^2 + y^2)y' = 2xy\).

Step 6: Match to Given Options

Compare the formatted differential equation \((x^2 + y^2)y' = 2xy\) to the options provided. It matches option (a).

Key Concepts

Implicit Differentiation

Implicit differentiation is a useful technique especially in dealing with equations where variables are not neatly separable. In such cases, like the equation of a circle, both dependent and independent variables (here, typically "y" and "x", respectively) are intertwined. By implicitly differentiating, we take the derivative of each side of the equation with respect to our independent variable, assuming that subsequent variables are functions of this independent variable.

- This means for an equation like, \(x^2 + y^2 - 2ay = 0\), differentiating with respect to \(x\) recognizes \(y\) as a function \(y(x)\). - Thus, terms involving \(y\) are differentiated using the chain rule. For instance, the term \(y^2\) becomes \(2y \cdot y'\), introducing \(y'\), the derivative of \(y\) with respect to \(x\).

The resulting expression we obtain after implicit differentiation helps us solve for \(y'\), allowing us a way to find the rate of change of \(y\) as \(x\) varies, given the implicit curve described initially.

Family of Circles

A family of circles refers to a collection of circles sharing common properties. In the given exercise, the equation \(x^2 + y^2 - 2ay = 0\) represents such a family.

- Here, the parameter \(a\) is a constant, which effectively dictates the position of the center of the circle along the y-value.- Variations in \(a\) shift the circle up or down without altering its shape, thereby altering the circle's specific identity within the family only.- All these circles are centered on the y-axis, demonstrating a common trait useful for deriving relationships and properties across the set.

This understanding of families of curves is crucial in differential equations as it aids in approaching problems by realizing they may share a single unifying differential equation, allowing for generalized analysis and problem-solving.

Coordinate Geometry

Coordinate geometry, commonly known as analytic geometry, marries algebra with geometry by using coordinate systems. This allows for the representation of geometrical shapes with algebraic equations.

- In this context, the circle equation \(x^2 + y^2 - 2ay = 0\) can be tackled by understanding its geometric meaning: circles are depicted as having a certain center and radius.- For our specific family of circles, as discussed, they all share a y-axis-central position but vary in where they rest on this axis, a result influenced by the constant \(a\).

Coordinate geometry provides a significant framework for visualizing and solving problems like this graphically. It connects the symbolic manipulations back to geometrical interpretations that are intuitively meaningful, enabling not just problem-solving on paper, but also visualization in space, making complex ideas accessible.