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MHT CET202619 April 2026Morning ShiftMathematicsDifferential EquationsActual

The solution of the differential equation dy dx = a + bx c + dy represents a family of circles centered at the origin if...

Options

  1. Aa = c = 0, b + d = 0
  2. Ba = c = 0, b = d
  3. Cb = d = 0, a + c = 0
  4. Db = d = 0, a = c

Correct answer

A. a = c = 0, b + d = 0

Step-by-step solution

The equation of a family of circles centered at the origin is given by x^2 + y^2 = r^2 , where r is an arbitrary constant. Differentiating this equation with respect to x , we get: 2x + 2y dy dx = 0 dy dx = - x y The given differential equation is: dy dx = a + bx c + dy For this to represent the family of circles centered at the origin, it must be identical to dy dx = - x y . Comparing the two equations, we get: a + bx c + dy = -x y This requires a = 0 and c = 0 . The equation then becomes: bx dy = - x y b d = -1 b

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