Question

The differential equation of system of concentric circles with center \((1,2)\) is ____

• A. \(\frac{dy}{dx}=\frac{x-1}{x-2}\)
• B. \(\frac{dy}{dx}=\frac{x-1}{2-x}\)
• C. \(\frac{dy}{dx}=\frac{1-x}{2-y}\)
• D. \(\frac{dy}{dx}=\frac{1-x}{y-2}\)

Hint:

Write the family of circles with the given center and eliminate the radius by differentiating once.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept
Concentric circles centered at \((1,2)\) have the equation \((x-1)^{2}+(y-2)^{2}=r^{2}\), with arbitrary \(r\).

Step 2: Differentiate
Differentiating once: \(2(x-1)+2(y-2)\frac{dy}{dx}=0\).

Step 3: Solve for dy/dx
\(\frac{dy}{dx}=-\frac{x-1}{y-2}=\frac{1-x}{y-2}\).

Final Answer:
The differential equation is \(\frac{dy}{dx}=\frac{1-x}{y-2}\). \[ \boxed{\text{(D) }\dfrac{dy}{dx}=\dfrac{1-x}{y-2}} \]