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AIEEE 2007

JEE Main 2007 Mathematics question on Differential Equations

The differential equation dydx=1−y2y\dfrac{dy}{dx} = \dfrac{\sqrt{1-y^2}}{y} determines a family of circles with
  1. A.
    variable radii and a fixed centre at (0,1)(0, 1)
  2. B.
    variable radii and a fixed centre at (0,−1)(0, -1)
  3. C.
    fixed radius 1 and variable centres along the XX-axis
  4. D.
    fixed radius 1 and variable centres along the YY-axis
Show answer and solution

Correct option: C

Solution

Separate variables: y dy/√(1-y²) = dx. Integrate both sides: -√(1-y²) = x + C, so √(1-y²) = -(x+C). Squaring: (x+C)² + y² = 1. This is a family of circles with radius 1 and centres at (-C, 0), which vary along the X-axis.

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