Question

Assertion (A): The order of the differential equation of a family of circles with constant radius is two.
Reason (R): An algebraic equation involving two arbitrary constants corresponds to the general solution of a second order differential equation.

• A. Both A and R are true, and R is the correct explanation of A
• B. A is true, R is false
• C. Both A and R are false
• D. A is false, R is true

Hint:

The number of arbitrary constants in a general solution equals the order of the differential equation. Differentiating removes constants step-by-step.

Answer 1

The Correct Option is A

The general equation of a circle with constant radius \( r \) is: \[ (x - a)^2 + (y - b)^2 = r^2 \] This has two arbitrary constants: \( a \) and \( b \) (the center coordinates).
Differentiating twice is required to eliminate both constants and obtain a differential equation, which makes the order 2.
So Assertion (A) is true.
Now, Reason (R) states:
> An algebraic equation having two arbitrary constants represents the general solution of a second-order differential equation.
This is correct and directly explains why the order is 2.
Thus, both A and R are true, and R is the correct explanation of A.