A curve amongst the family of curves represented by the differential equation, x 2 - y 2   d x + 2 x y…

A curve amongst the family of curves represented by the differential equation, x2-y2 dx+2xy dy=0 which passes through 1,1, is
  1. A circle with centre on the x- axis.
  2. A circle with centre on the y- axis.
  3. A hyperbola with transverse axis along the x- axis.
  4. An ellipse with major axis along the y- axis.

Solution

x2dx-y2dx+2xy dy=0

x2dx=y2dx-2x ydy

⇒-dx=dy2x  (exact differential equation)

⇒ -x=y2x+C

⇒ x2+y2+Cx=0

∵ It passes through 1,1.

Hence, C=-2

⇒ x2+y2-2x=0

Asked in: JEE Main 2019 (10 Jan Shift 2)

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