Question

 The solution of the differential equation \[ (x-1)\frac{dx}{dy}+(y-2)=0, \] given that \(x=1\) when \(y=1\), represents a: 

• A. Parabola
• B. Circle
• C. Ellipse
• D. Hyperbola

Hint:

The general form \( (x-h)^2 + (y-k)^2 = r^2 \) represents a circle.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
This is a separable differential equation. We rearrange the terms to isolate \( x \) and \( y \), then integrate both sides.

Step 2: Detailed Explanation:
Given \( (x - 1) \frac{dx}{dy} = -(y - 2) \).
Separate the variables: \( (x - 1) dx = -(y - 2) dy \).
Integrate both sides: \( \int (x - 1) dx = \int -(y - 2) dy \).
\( \frac{(x - 1)^2}{2} = -\frac{(y - 2)^2}{2} + C \).
Multiply by 2: \( (x - 1)^2 + (y - 2)^2 = 2C \).
Using the initial condition \( x = 1, y = 1 \): \( (1 - 1)^2 + (1 - 2)^2 = 2C \implies 0 + 1 = 2C \implies 2C = 1 \).
The equation is \( (x - 1)^2 + (y - 2)^2 = 1 \).

Step 3: Final Answer:
This is the equation of a circle centered at \((1, 2)\) with radius \( 1 \).