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A circle passes through the origin and cuts orthogonally each of the circles x² + y² - 8y + 12 = 0 and x² + y² - 4x - 6y - 3 = 0. If R is the radius of this circle, find the value of 2R / sqrt(5).

1. 1
2. 2
3. 3
4. 4

Correct answer: 3

Solution

Let the required circle be x²+y²+2gx+2fy+c=0. It passes through origin => c=0. Orthogonality with circle 1 (g1=0, f1=-4, c1=12): 2g(0)+2f(-4) = c+12 => -8f = 12 => f = -3/2. Orthogonality with circle 2 (g2=-2, f2=-3, c2=-3): 2g(-2)+2f(-3) = c+(-3) => -4g-6f = -3 => -4g-6(-3/2)=-3 => -4g+9=-3 => g=-3. Radius R = sqrt(g²+f²-c) = sqrt(9+9/4) = sqrt(45/4) = 3sqrt(5)/2. 2R/sqrt(5) = 2*(3sqrt(5)/2)/sqrt(5) = 3.

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