Question

The differential equation of the family of lines having \(x\)-intercept \(a\) and \(y\)-intercept \(b\) is

• A. \(\dfrac{d^2y}{dx^2} = -1\)
• B. \(\dfrac{d^2y}{dx^2} = 10\)
• C. \(\dfrac{d^2y}{dx^2} = 1\)
• D. \(\dfrac{d^2y}{dx^2} = 0\)

Hint:

Any family of straight lines always leads to \(\dfrac{d^2y}{dx^2}=0\).

Answer 1

The Correct Option is D

Step 1: Write the equation of the family of lines.
A line with \(x\)-intercept \(a\) and \(y\)-intercept \(b\) is \[ \frac{x}{a} + \frac{y}{b} = 1 \]
Step 2: Rewrite in explicit form.
\[ y = b - \frac{b}{a}x \]
Step 3: Differentiate w.r.t. \(x\).
\[ \frac{dy}{dx} = -\frac{b}{a} = \text{constant} \]
Step 4: Differentiate again.
\[ \frac{d^2y}{dx^2} = 0 \]