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MHT CET202519 Apr 2025Evening ShiftMathematicsApplication of DerivativesActual

A normal is drawn at a point P (x, y) of a curve y = f (x) . The normal meets the X axis at Q . l( PQ )= k .( k is a constant ) Then equation of the curve through (0, k ) is

Options

  1. Ax^2+ y ^2= k ^2
  2. B(1+ k ) x^2+ y ^2= k ^2
  3. Cx^2+ (1+ k ^2 ) y ^2= k ^2
  4. Dx^2+2 y ^2=2 k ^2

Correct answer

A. x^2+ y ^2= k ^2

Step-by-step solution

Find the family of curves where the length of the normal from any point to the x-axis is constant. Consider a point P (x, y) on the curve. The slope of the tangent at P is dy dx , so the slope of the normal is - dx dy . The equation of the normal is Y - y = - dx dy ( X - x) . It intersects the x-axis at Q , where Y = 0 . Substituting gives: 0 - y = - dx dy ( X _Q - x) , which implies X _Q = x + y dy dx . Given that length PQ = k , apply the distance formula: ( PQ )^2 = ( x + y dy dx - x )^2 + (0 - y)^2 = k^2 . Simp

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