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A point P moves in the plane so that the sum of the squares of its…
Question
A point P moves in the plane so that the sum of the squares of its distances from the three fixed points A(1,2), B(5,2) and C(3,6) is constant and equal to 60. The locus of P is a circle. If the square of its radius equals m/n in lowest terms, find m+n.
✓ Verified answer: 133checked by our engine — not a guess
Step-by-step solution
Let P = (x,y). Then
PA^2 + PB^2 + PC^2 = (x-1)^2+(y-2)^2 + (x-5)^2+(y-2)^2 + (x-3)^2+(y-6)^2 = 60.
Expand: 3x^2 - (2+10+6)x + (1+25+9) + 3y^2 - (4+4+12)y + (4+4+36) = 60
=> 3x^2 - 18x + 35 + 3y^2 - 20y + 44 = 60
=> 3x^2 + 3y^2 - 18x - 20y + 79 = 60
=> 3x^2 + 3y^2 - 18x - 20y + 19 = 0.
Divide by 3: x^2 + y^2 - 6x - (20/3)y + 19/3 = 0.
Centre = (3, 10/3). Radius^2 = 3^2 + (10/3)^2 - 19/3 = 9 + 100/9 - 57/9 = 81/9 + 100/9 - 57/9 = 124/9.
m/n = 124/9, m+n = 133.
Final answer133
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