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

We are given a family of straight lines $y = mx$, where $m$ is a parameter (arbitrary constant).
To find the differential equation, differentiate both sides:
$\dfrac{dy}{dx} = m$
But from the original equation, $m = \dfrac{y}{x}$
So, $\dfrac{dy}{dx} = \dfrac{y}{x} \Rightarrow x \dfrac{dy}{dx} - y = 0$
Thus, the Assertion is true.
Now, Reason (R) states the general method of obtaining a differential equation — by differentiating and eliminating arbitrary constants.
That is exactly what was done here.
So both Assertion and Reason are true, and Reason correctly explains Assertion.