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AP EAMCET2003MathematicsDifferential Equations

The differential equation of the family of parabola with focus as the origin and the axis as X -axis, is :

Options

  1. Ay ( d y d x )^2+4 x d y d x =4 y
  2. B-y ( d y d x )^2=2 x d y d x -y
  3. Cy ( d y d x )^2+y=2 x y d y d x
  4. Dy ( d y d x )^2+2 x y d y d x +y=0

Correct answer

B. -y ( d y d x )^2=2 x d y d x -y

Step-by-step solution

Given that, Focus S=(0,0) , let P(x, y) be any point on the parabola. Since, S P^2=P M^2 (x-0)^2+(y-0)^2=(x+a)^2 x^2+y^2=x^2+a^2+2 x a y^2=2 a x+a^2 (i) 2 y d y d x =2 a (ii) From Eqs. (i) and (ii), we get y^2=2 y d y d x x+ (y d y d x )^2 y^2 ( d y d x )^2+2 x y d y d x =y^2 y ( d y d x )^2+2 x d y d x =y -y ( d y d x )^2=2 x d y d x -y

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