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AP EAMCET202420 May 2024Evening ShiftMathematicsApplication of DerivativesActual

The acute angle between the curves x^2+y^2=x+y and x^2+y^2=2 y is

Options

  1. A2 3
  2. B2
  3. C3
  4. D4

Correct answer

D. 4

Step-by-step solution

x^2+y^2=x+y .... (i) x^2+y^2=2 y .... (ii) Add (i) and (ii), 2 x^2+2 y^2-3 y-x=0 .... (iii) Subtract (i) from (ii), x+y=0 .... (iv) Differentiate (i) w.r.t. x d y d x = 2 x-1 1-2 y =m₁ Differentiate (ii) w.r.t. x d y d x = x y = Angle between (i) and (ii) is given by aligned & = | m₁-m₂ 1+m₁ m₂ |= | 2 x-1 1-2 y - x 1-y 1+ (2 x-1) x (1-2 y)(1-y) | & = | (2 x-1)(1-y)-x(1-2 y) (1-2 y)(1-y)+(2 x-1) x | & = | x+y-1 2 x^2+2 y^2-3 y-x+1 | aligned Using (iii) and (iv) =|-1| = 4

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