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AP EAMCET202315 May 2023Morning ShiftMathematicsCircleActual

Let the circle S x^2+y^2+2 g x+2 f y+c=0 cut the circles x^2+y^2-2 x+2 y-2=0 and x^2+y^2+4 x-6 y+9=0 orthogonally. If the centre of the circle S=0 lies on the line 2 x +3 y -2=0 , then 2 ~g + f =

Options

  1. Ac
  2. Bc+f
  3. C2 ~g - c
  4. Dc-f

Correct answer

D. c-f

Step-by-step solution

In circle S=x^2+y^2+2 g x+2 f y+c=0...(1) centre =(-g,-f), g_ i =g, f₁=f, c₁=c In circle x^2+y^2-2 x+2 y-2=0...(2) g₂=-1, f₂=1, c₂=-2 In circle x^2+y^2+4 x-6 y+9=0...(3) g₃=2, f₃=-3, c₃=9 Since eq (1) & (2) are orthogonal hence aligned & 2 g₁ g₂+2 f₁ f₂=c₁+c₂ & -2 g+2 f=c-2...(4) aligned Since eq (1) & (3) are also orthogonal Hence 4 g-6 f=c+9...(5) Since center (-g,-f) lies on 2 x+3 y-2=0 Hence -2 g-3 f-2=0 Solving eq (4), (5) and (6), we get g= 1 2 , f=-1, c=-1 Therefore 2 g+f=0=c-f

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