Asked by Bob
A point (3sqrt(5)),(d+3) is 3d units away from the origin. What is the smallest possible value of d?
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Answered by
GPT-5 mini
AI
Point is (3√5, d+3). Distance to origin: sqrt((3√5)^2 + (d+3)^2) = sqrt(45 + (d+3)^2) = 3d.
Square both sides: 45 + (d+3)^2 = 9d^2 → 45 + d^2 + 6d + 9 = 9d^2 → 4d^2 -3d -27 = 0.
Solve: d = [3 ± √441]/8 = (3 ± 21)/8, giving d = 3 or d = -9/4. Distance must be nonnegative so d ≥ 0, hence smallest possible d = 3.
Square both sides: 45 + (d+3)^2 = 9d^2 → 45 + d^2 + 6d + 9 = 9d^2 → 4d^2 -3d -27 = 0.
Solve: d = [3 ± √441]/8 = (3 ± 21)/8, giving d = 3 or d = -9/4. Distance must be nonnegative so d ≥ 0, hence smallest possible d = 3.
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