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NTA Abhyas JEE Main2020MathematicsDifferential EquationsPractice

The differential equation of the family of curves whose tangent at any point makes an angle of π 4 with the ellipse x 2 4 + y 2 = 1 is

Options

  1. Ad y d x = x + y x - y
  2. Bd y d x = x + 4 y x - 4 y
  3. Cd y d x = x 4 y
  4. Dd y d x = 4 y x

Correct answer

B. d y d x = x + 4 y x - 4 y

Step-by-step solution

Given, x 2 4 + y 2 = 1 On differentiating w.r.t. x , x 2 + 2 y d y d x = 0 d y d x = - x 4 y = m 1   l e t Let, the slope of the tangent at a point on ellipse where the tangents meet is m 2 Given, t a n π 4 = m 1 - m 2 1 + m 1 m 2 1 = - x 4 y - m 2 1 + x 4 y m 2 ⇒ 4 y - x m 2 = x + 4 y m 2 4 y - x m 2 = x + 4 y m 2 or 4 y - x m 2 = - x - 4 y m 2 m 2 = 4 y - x x + 4 y or m 2 = 4 y + x x - 4 y d y d x = 4 y - x x + 4 y or d y d x = 4 y + x x - 4 y

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