Question

Assertion (A): The differential equation representing the family of curves $y = mx$, $m$ being an arbitrary constant, is \[ x \dfrac{dy}{dx} - y = 0 \]
Reason (R): For a family of curves, the differential equation is obtained by differentiating the equation of family with respect to $x$ and then eliminating the arbitrary constant, if any.

• A. Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A)
• B. Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation for Assertion (A)
• C. Assertion (A) is true, but Reason (R) is false
• D. Assertion (A) is false, but Reason (R) is true

Hint:

To derive a differential equation from a family of curves, differentiate and eliminate the arbitrary constant.

Answer 1

The Correct Option is A

The family of straight lines is given by $y = mx$, where $m$ is an arbitrary constant parameter.
To derive the differential equation, differentiate the equation with respect to $x$: $ \dfrac{dy}{dx} = m $
From the original equation, we can express $m$ as $m = \dfrac{y}{x}$.
Substituting this into the differentiated equation gives: $\dfrac{dy}{dx} = \dfrac{y}{x}$.
Rearranging this equation yields: $x \dfrac{dy}{dx} - y = 0$.
Therefore, the Assertion is true.
The Reason (R) describes the general procedure for forming a differential equation: differentiation and elimination of arbitrary constants.
This procedure was precisely followed in deriving the differential equation for the given family of lines.
Consequently, both the Assertion and the Reason are true, and the Reason accurately explains the Assertion.