Question

Assertion (A): Order of the differential equations of a family of circles with constant radius is two.
Reason (R): An algebraic equation having two arbitrary constants is general solution of a \(2^{\text{nd}}\) order differential equation.

• (A) and (R) are true, (R) is the correct explanation to (A)
• (A) is true, (R) is false
• (A) and (R) are false, (R) is not the correct explanation to (A)
• (A) is false, (R) is true

Hint:

The order of a differential equation obtained from a family of curves equals the number of arbitrary constants present in the family.

Answer 1

The Correct Option is A

Step 1: Write the family of circles with constant radius.
A family of circles having constant radius \(r\) can be written as \[ (x-a)^2+(y-b)^2=r^2 \] where \[ a \quad \text{and} \quad b \] are arbitrary constants representing the center of the circle.
Thus, the family contains two arbitrary constants.

Step 2: Determine the order of the differential equation.
To obtain the differential equation, we eliminate the arbitrary constants \(a\) and \(b\).
Since there are two arbitrary constants, we need two differentiations.
Therefore, the resulting differential equation is of order \(2\).
Hence, Assertion (A) is true.

Step 3: Examine the Reason (R).
The statement: “An algebraic equation having two arbitrary constants is the general solution of a second order differential equation.” is true because the order of a differential equation is equal to the number of independent arbitrary constants present in its general solution.
Hence, Reason (R) is also true.

Step 4: Check whether R explains A.
The assertion is directly based on the fact stated in the reason. Since the family of circles has two arbitrary constants, its differential equation is of second order.
Thus, Reason (R) correctly explains Assertion (A).

Step 5: Final conclusion.
Therefore, \[ \boxed{\text{(A) and (R) are true, and (R) is the correct explanation to (A)}} \]