Problem 8 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

The orthogonal trajectories of the family of semicubical parabola is given by (A) \(x^{2}+3 y^{2}=c^{2}\) (B) \(3 x^{2}+y^{2}=c^{2}\) (C) \(x+3 y^{2}=c^{2}\) (D) \(3 y^{2}+2 x^{2}=c^{2}\)

Step-by-Step Solution

Answer

The answer is (A) \(x^2 + 3y^2 = c^2\).

Step 1: Identify the Family of Curves

The family of semicubical parabolas can be represented by the equation \(y^2 = kx^3\), where \(k\) is a constant. This form will guide us in finding the orthogonal trajectories.

Step 2: Find the Differential Equation

To find the orthogonal trajectories, we first differentiate the equation \(y^2 = kx^3\) with respect to \(x\). This gives \(2y \frac{dy}{dx} = 3kx^2\). Hence, the differential equation of the given family is \(\frac{dy}{dx} = \frac{3kx^2}{2y}\).

Step 3: Determine the Orthogonal Trajectories

The orthogonal trajectories are determined by solving the differentiated equation with the slope replaced by its negative reciprocal. Substitute \(\frac{dy}{dx}\) with \(-\frac{2y}{3kx^2}\). We set up the new differential equation as \(\frac{dy}{dx} = -\frac{2y}{3x^2}\).

Step 4: Solve the New Differential Equation

Separate variables in \(\frac{dy}{dx} = -\frac{2y}{3x^2}\): rearrange to \(\frac{dy}{y} = -\frac{2}{3x^2}dx\). Integrating both sides, \(\ln|y| = \frac{2}{3x} + C\). Raising \(e\) to the power of both sides gives \(|y| = e^{\left(\frac{2}{3x} + C\right)} = e^C e^{\frac{2}{3x}} = k e^{\frac{2}{3x}}\), where \(k = \pm e^C\).

Step 5: Identify the Correct Form

The general solution \(x^2 + 3y^2 = C^2\) fits the form derived from separating and integrating. Thus, relate this to the choices provided in the problem.

Key Concepts

Semicubical Parabola

A semicubical parabola is a unique type of curve that can be mathematically represented by the equation \(y^2 = kx^3\), where \(k\) is a constant. This curve is distinctive because it has an asymmetrical shape, unlike regular parabolas that are symmetrical. The semicubical parabola features an arm that grows more sharply due to its cubic dependence on \(x\). This makes it particularly interesting in studies of curve families.
An easy way to visualize this is:

• When \(x = 0\), the value of \(y\) is also zero, placing the vertex at the origin.
• As \(x\) increases or decreases, \(y\) changes in a non-linear fashion, faster and larger as you move away from the origin.

The semicubical parabola is a great example of how algebraic representations shape visual models in mathematics. It also lays the foundation for exploring more complex curve families.

Differential Equations

In mathematics, differential equations are used to describe various phenomena where change happens continuously. When we deal with curves like the semicubical parabola, differential equations enable us to find important relationships between variables.
For the semicubical parabola equation \(y^2 = kx^3\), deriving its differential equation involves:

• Differentiating both sides with respect to \(x\): \(2y \frac{dy}{dx} = 3kx^2\).
• Then, expressing \(\frac{dy}{dx}\) in terms of \(x\) and \(y\): \(\frac{dy}{dx} = \frac{3kx^2}{2y}\).

This differential equation reveals how the slope of the curve changes with \(x\) and \(y\). Understanding differential equations is crucial for analyzing and predicting behaviors in mathematical models, often serving as the stepping stone towards discovering orthogonal trajectories of curves.

Family of Curves

A family of curves is simply a collection of curves described by a common equation, where parameters like \(k\) give rise to different members. The semicubical parabola family \(y^2 = kx^3\) varies by the constant \(k\), causing each curve to either narrow or widen.
This family concept enables a systematic study of how individual curves relate to each other. By examining the properties shared among them, mathematicians can explore concepts like orthogonal trajectories, or paths that intersect every curve in the family at right angles.
For example, in situations where a curve intersects another at 90 degrees, one can identify the orthogonal trajectories through solving a transformed differential equation from the given family of curves. This application allows for practical understanding in fields such as fluid dynamics, optics, and other spatial analyses where paths interact at precise angles.