Question

The differential equation of all lines where the length of the normal from the origin is p and the inclination of the normal is \(α\) is... (where p and \(α\) are arbitrary constants)

• A. \(\frac{d^2y}{dx^2} = 0\)
• B. \(\frac{dy}{dx} = 0\)
• C. \(\frac{dy}{dx} = -cotα\)
• D. \(\frac{d^2y}{dx^2} = csc^2α\)

Hint:

The family has two arbitrary constants, so the equation is of second order.

Answer 1

The Correct Option is A

Step 1: Count parameters
The data are $p$ and $\alpha$, both arbitrary. The order of the differential equation equals the number of independent arbitrary constants, which is 2.

Step 2: Describe the family
Every non-vertical straight line can be written as $y = mx + c$ using suitable $p$ and $\alpha$.

Step 3: Differentiate twice
$y' = m$, $y'' = 0$. The constants vanish together.

Step 4: Conclusion
The required differential equation is $\frac{d^2y}{dx^2} = 0$, option (A).

Final Answer:
The equation is d^2y/dx^2 = 0. This is option (A). \[ \boxed{\text{(A) }\frac{d^2y}{dx^2}=0} \]