Question

If the solution of the differential equation \( \frac{dy}{dx} = ax + 32y + f \) represents a circle, then the value of \( a \) is

• A. 2
• B. -2
• C. 3
• D. -4

Hint:

If the solution of the differential equation $dydx=\fracax+3/2y+f$ represents a circle, then the value of a is

Answer 1

The Correct Option is B

Step 1: Concept
A general second-degree equation represents a circle if the coefficients of $x^2$ and $y^2$ are equal and the $xy$ term is absent.
Step 2: Analysis
Cross-multiplying: $(2y+f)dy = (ax+3)dx$.
Step 3: Evaluation
Integrating both sides: $y^2 + fy = a\frac{x^2}{2} + 3x + c \Rightarrow -\frac{a}{2}x^2 + y^2 - 3x + fy - c = 0$.
Step 4: Conclusion
For this to be a circle, the coefficient of $x^2$ must be 1 (same as $y^2$). So, $-a/2 = 1 \Rightarrow a = -2$.
Final Answer: (b)