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COMEDK2020MathematicsCircle

If the circles x²+y²-2 x-2 y-7=0 and x²+y²+4 x+2 y+k=0 cut orthogonally, then the length of the common chord of the circles is

Options

  1. A2
  2. B12 / 13
  3. C8
  4. D5

Correct answer

B. 12 / 13

Step-by-step solution

We have, aligned &S₁: x²+y²-2 x-2 y-7=0 &S₂: x²+y²+4 x+2 y+k=0 aligned Since, S₁ and S₂ cut orthogonally, then aligned & & 2 g₁ E₂+2 f₁ f₂ &=c₁+c₂ & 2(-1)(2)+2(-1)(1) &=-7+k &-4-2 &=-7+k k^ &=1 aligned Now, let r₁ and r₂ be the radius of S ₁ and S ₂ , respectively. Then, aligned &n₁= (-1)²+(-1)²-(-7) =3 &r₂= (2)²+(1)²-1 =2 aligned Now, length of common chord aligned &= 2 r₂ r₁²+r₂² &= 2 3 2 (3)²+(2)² = 12 13 aligned

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