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MHT CET202523 Apr 2025Morning ShiftMathematicsDifferential EquationsActual

The differential equation of all circles touching the Y-axis at the origin and centre on the X -axis is

Options

  1. Ax^2+ y ^2+2 x y dy d x =0
  2. Bx^2-y^2+2 x y d y d x =0
  3. C2 x^2+ y ^2+x y dy d x =0
  4. Dx^2-2 y ^2+2 x y dy d x =0

Correct answer

B. x^2-y^2+2 x y d y d x =0

Step-by-step solution

A circle touching the Y -axis at the origin has its center on the X -axis, located at (a,0) , with radius r = |a| . Its equation is (x - a)^2 + y^2 = a^2 , which simplifies to: x^2 + y^2 - 2ax = 0 Differentiating both sides with respect to x gives: 2x + 2y dy dx - 2a = 0 Dividing by 2 and rearranging yields: a = x + y dy dx Substituting this into the original equation results in: x^2 + y^2 - 2x (x + y dy dx ) = 0 Simplifying the expression leads to: x^2 - y^2 - 2xy dy dx = 0 This corresponds to option B. The correc

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