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AP EAMCET2016MathematicsCircle

If the angle between the circles x^2+y^2-2 x-4 y+c=0 and x^2+y^2-4 x-2 y+4=0 is 60^ , then c is equal to

Options

  1. A3 5 2
  2. B6 5 2
  3. C9 5 2
  4. D7 5 2

Correct answer

D. 7 5 2

Step-by-step solution

We have, aligned & C ₁: x^2+y^2-2 x-4 y+c=0 ... (i) & C ₂: x^2+y^2-4 x-2 y+4=0... (ii) aligned So, centre of circle C ₁=(1,2), r₁= 5-c and centre of circle C ₂=(2,1), r₂=1 Now, angle between the two circles, aligned & = ( C ₁ C ₂ )^2- (r₁^2+r₂^2 ) 2 r₁ r₂ & = [(2-1)^2+(1-2)^2 ]-(5-c+1) 2 5 -c(1) aligned Given, =60^ aligned & 60^ = 1+1-5+c-1 2 5-c & 1 2 = c-4 2 5-c & 5-c =c-4 aligned On squaring both sides, we get aligned & 5-c=c^2-8 c+16 & c^2-5 c+11=0 & c= 7 49-44 2 & c= 7 5 2 aligned

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