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MHT CET202521 Apr 2025Evening ShiftMathematicsDifferential EquationsActual

The differential equation of all circles having their centres on the line y =5 and touching X -axis is ......

Options

  1. A(5- y ) d y ~d x + y ^2-10 y =0
  2. B(5- y )^2 ~d ^2 y ~d x^2 + y ^2-10 y =0
  3. C(5- y ) dy d x + y -10=0
  4. D(5- y )^2 ( dy d x )^2+ y ^2-10 y =0

Correct answer

D. (5- y )^2 ( dy d x )^2+ y ^2-10 y =0

Step-by-step solution

A circle with center on the line y = 5 and tangent to the x-axis has radius r = |5| = 5 , yielding the equation (x - h)^2 + (y - 5)^2 = 25 . Differentiating both sides with respect to x gives 2(x - h) + 2(y - 5) dy dx = 0 , which simplifies to (x - h) = -(y - 5) dy dx . Substituting this back into the original equation produces [(y - 5) dy dx ]^2 + (y - 5)^2 = 25 , or equivalently (y - 5)^2( dy dx )^2 + y^2 - 10y + 25 = 25 . Subtracting 25 from both sides results in (y - 5)^2( dy dx )^2 + y^2 - 10y = 0 , matching o

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