Problem 89 - A Complete Resource Book in Mathematics for JEE Main

Question

A curve which cuts every member of a given family of curves according to a given law is called a trajectory of the family. We shall consider only the case when each trajectory cuts every member of a given family at a constant angle. The trajectory will be called Orthogonal if the constant angle is a right angle. For example, every line through the origin of coordinates is an orthogonal trajectory of the family of concentric circles with centre at the origin. Let the equation of the given family of curves be $$ f(x, y, c)=0 $$ Differentiate (1) and eliminate the arbitaray constant \(c\) between (1) and the resulting equation. That gives the differential equation of the family (1). Let it be \(F\left(x, y, \frac{d y}{d x}\right)=0\) Replace \(\frac{d y}{d x}\) by \(-\frac{d x}{d y}\) The differential equation of the orthogonal trajectory is $$ F\left(x, y,-\frac{d x}{d y}\right)=0 $$ Integrate (3) to get the equation of the required orthogonal trajectory. Orthogonal trajectory of the family of hyperbolas \(x y=k^{2}\) is (A) \(x^{2}+y^{2}=c\) (B) \(x^{2}-y^{2}=c\) (C) \(2 x^{2}-2 y^{2}=c\) (D) None of these

Step-by-Step Solution

Answer

The orthogonal trajectory is \( x^2 + y^2 = c \), answer (A).

Step 1: Differentiate the given family equation

Given the family of hyperbolas: \( xy = k^2 \), differentiate implicitly with respect to \( x \): \( y + x \frac{dy}{dx} = 0 \).

Step 2: Solve for dy/dx

From the differentiated equation, solve for \( \frac{dy}{dx} \): \( \frac{dy}{dx} = -\frac{y}{x} \). This is the slope of the given family of curves.

Step 3: Find the slope of orthogonal trajectories

Replace \( \frac{dy}{dx} \) with \( -\frac{dx}{dy} \) to find the orthogonal slope: \( \frac{dx}{dy} = \frac{x}{y} \).

Step 4: Rearrange the orthogonal slope equation

Rearrange \( \frac{dx}{dy} = \frac{x}{y} \) to \( y \cdot dx = x \cdot dy \), which integrates more easily.

Step 5: Integrate to find orthogonal trajectories

Integrate: \( \int y \, dx = \int x \, dy \), resulting in \( yx = \frac{x^2}{2} + C \). Rearrange to \( x^2 + y^2 = C' \) by completing the square.

Step 6: Complete the equation of orthogonal trajectories

The equation \( x^2 + y^2 = C' \) represents circles centered at the origin, which matches the form \( x^2 + y^2 = c \).

Key Concepts

Differential Equations

Differential equations are equations that express a relationship between a function and its derivatives. They are essential in describing various phenomena such as motion, heat, and waves. In the context of orthogonal trajectories, differential equations help identify curves that intersect a given family of curves at specific angles, such as 90 degrees (orthogonal).
The key idea is to find a new curve that cuts across a set of curves in a consistent manner, according to some rule or angle.
To begin with, we have the original family of curves represented by a differential equation. Here, for example, the hyperbola family is shown by \( xy = k^2 \). By differentiating this equation implicitly, we find a slope expression \( \frac{dy}{dx} = -\frac{y}{x} \). This slope represents the direction each curve in the family takes at a given point.
To determine the orthogonal trajectories—curves that intersect the original family perpendicularly—we swap the slope, using the negative reciprocal \( \frac{dx}{dy} = \frac{x}{y} \). Solving this results in a new differential equation that can be integrated to find the orthogonal trajectory.

Implicit Differentiation

Implicit differentiation is a technique used to differentiate equations where the variables \( x \) and \( y \) are not easily separated. This method is particularly useful when dealing with equations that define a family of curves and their relationships, as seen in our hyperbola example.
By differentiating expression like \( xy = k^2 \) implicitly, we aim to express the derivative \( \frac{dy}{dx} \) without having to explicitly solve \( y \) in terms of \( x \).

• Differentiating \( xy = k^2 \) with respect to \( x \), for instance, requires applying the product rule to obtain \( y + x \frac{dy}{dx} = 0 \).
• From this, we derive \( \frac{dy}{dx} = -\frac{y}{x} \), showing how changes in \( x \) and \( y \) interact along the curve.

This approach is essential when solving problems involving orthogonal trajectories, as it simplifies the process of finding derivatives when dealing with complex relationships in the family of curves.

Family of Curves

A family of curves is a collection of related curves that share a common mathematical property, represented by an equation with a parameter. This concept is fundamental when exploring orthogonal trajectories, as these curves intersect each member of a family following a specific pattern.
In problems where orthogonal trajectories are involved, we're given a family of curves, typically described by an equation like \( xy = k^2 \), which represents hyperbolas having different sizes depending on the parameter \( k \).

• Each member of this family intersects at certain points, defined by their individual equations.
• Finding the orthogonal trajectories requires understanding the "shape" of this family and how each curve behaves.

Once we implement the differential equation approach, we find a new set of equations representing trajectories that cut across the family at right angles, such as the circles \( x^2 + y^2 = c \) intersecting the hyperbolas. This distinctive relationship gives insights into how curves relate geometrically in the plane.