Problem 97 - A Complete Resource Book in Mathematics for JEE Main

Question

In the following questions an Assertion \((A)\) is given followed by a Reason \((R) .\) Mark your responses from the following options: (A) Assertion(A) is True and Reason(R) is True; Reason(R) is acorrect explanation forAssertion(A) (B) Assertion(A) is True, Reason(R) is True; Reason(R) is not a correct explanation for Assertion(A) (C) Assertion(A) is True, Reason(R) is False (D) Assertion(A) is False, Reason(R) is True Assertion: The differential equation of all straight lines which are at a constant distance \(p\) from the origin is \(\left(y-x y_{1}\right)^{2}=p^{2}\left(1+y_{1}^{2}\right)\) Reason: The general equation of any straight line which is at a constant distance \(p\) from the origin is \(x \cos \alpha+y \sin \alpha=p .\)

Step-by-Step Solution

Answer

(A) Assertion(A) is True and Reason(R) is True; Reason(R) is a correct explanation for Assertion(A).

Step 1: Analyze the Assertion

The given differential equation is \((y-xy_1)^2 = p^2(1+y_1^2)\). This equation is supposed to represent all straight lines at a constant distance \(p\) from the origin. We need to verify if this assertion holds.

Step 2: Understand the General Equation for Lines

The general equation of a line at a constant distance \(p\) from the origin is \(x \cos \alpha + y \sin \alpha = p\), where \(\alpha\) is the angle made by the normal to the line with the x-axis. This equation indeed describes all lines at a constant distance from the origin.

Step 3: Relate General Equation to Given Differential Equation

The differential equation \((y-xy_1)^2 = p^2(1+y_1^2)\) needs to be checked against the general form \(x \cos \alpha + y \sin \alpha = p\). Consider the slope \(y_1 = \frac{dy}{dx}\) of the line, and check if substituting \(y_1\) into the general form results in the given differential equation. This differential form should ideally match the structure of the translated equation, confirming the Assertion.

Step 4: Validate the Relationship

After substituting \(y_1\) into the general form, the Assertion can be derived from the general equation \(x \cos \alpha + y \sin \alpha = p\), making both the Assertion and the Reason true.

Step 5: Determine Correct Answer

Since the Reason provides the same lines that the Assertion describes through its differential form, and both are mathematically interlinked, the Reason correctly explains the Assertion.

Key Concepts

Assertion and Reason

In mathematics, particularly in problems related to logical reasoning, an assertion is a statement proposed to be true, and a reason follows it to either support or explain the assertion.
When dealing with assertions and reasons, we usually have four possible relationships between them:

• (A) Both the Assertion and Reason are true, and the Reason correctly explains the Assertion.
• (B) Both are true, but the Reason does not explain the Assertion.
• (C) The Assertion is true, but the Reason is false.
• (D) The Assertion is false, but the Reason is true.

Understanding these relationships helps in logically deducing correct answers in exercises related to differential equations. In our exercise's context, the assertion involves the differential equation representing lines a constant distance from the origin, while the reason involves the general equation of such lines.

General Equation of a Line

The general equation of a line describes its orientation and specific position in a 2D plane.
One common form of the line equation relates it to its normal, or perpendicular distance, from the origin. This equation is given by: \[x \cos \alpha + y \sin \alpha = p\] where \(\alpha\) is the angle between the normal of the line and the x-axis, and \(p\) is the constant distance from the origin.

• \(\cos\alpha\) and \(\sin\alpha\) represent the direction cosines of the line's normal vector.
• Using this form is useful for lines whose orientation is known relative to the origin.

In the exercise, this form was used to establish a connection between the differential form and distance-based description of a line from the origin.

Distance from Origin

Calculating the distance of a geometric object like a line from the origin is an important concept in coordinate geometry.
For a straight line, this distance can help derive alternate forms of the line’s equation. The general equation \[x \cos \alpha + y \sin \alpha = p\] helps in defining the distance from the origin, which is represented as \(p\).

• In this context, \(p\) is a measure of how far the line is from the point (0,0) on the plane.
• This parameter becomes useful in the assertion-reason exercise to connect the differential equation with known geometric principles.

Using the distance from origin formula in equations makes it easier to identify and analyze lines in geometry and mathematics.

Slope of a Line

The slope is a critical component in understanding linear equations and geometry. It denotes the steepness and direction of a line in the coordinate plane.
Mathematically, slope \(m\) is expressed as \(m = \frac{rise}{run} = \frac{dy}{dx}\).
In the given problem, the use of slope \(y_1 = \frac{dy}{dx}\) allows transitioning between different forms of straight line equations:

• It helps relate the general equation of line with its differential equation counterpart.
• By substituting \(y_1\), we analytically match the assertion form with its alternative standard form.

Understanding the slope is essential not only for solving geometric problems but for linking various formats of linear expressions, aiding in both analytical and practical applications.