Question

The differential equation satisfied by circles with radius 3 and center lying on the Y-axis is

• A. $(\frac{dy}{dx} )^2 = ( \frac{x^2}{ 9 + x^2})$
• B. $(\frac{dy}{dx} )^2= ( \frac{ 9 + y^2}{y^2})$
• C. $(\frac{dy}{dx} )^2= ( \frac{x^2}{ 9 - x^2})$
• D. $(\frac{dy}{dx}^2 )= ( \frac{ 9 - y^2}{y^2})$

Answer 1

The Correct Option is C

Equation of a Circle and Its Derivative

Step 1: Given Equation of the Circle

The equation of a circle with radius r = 3 centered at the Y-axis is: 

x² + y² = 9

Step 2: Differentiating Both Sides

Differentiating implicitly with respect to x:

2x + 2y (dy/dx) = 0

Step 3: Solving for dy/dx

Rearranging:

dy/dx = -x / y

Step 4: Squaring Both Sides

Squaring both sides:

(dy/dx)² = x² / (9 - x²)

Step 5: Conclusion

This matches option (C), confirming that the derivative is correctly calculated.