Question

Two particles P and Q located at the points \( P(t, t^3 - 16t - 3) \), \( Q(t+1, t^3 - 6t - 6) \) are moving in a plane. The minimum distance between the points in their motion is:

• A. 1
• B. 5
• C. 169
• D. 49

Hint:

Use the distance formula between moving points and minimize using calculus or algebraic simplification.

Answer 1

The Correct Option is A

Let \( P = (t, t^3 - 16t - 3) \), \( Q = (t+1, t^3 - 6t - 6) \). The distance between P and Q is \[ D(t) = \sqrt{(1)^2 + [(t^3 - 6t - 6) - (t^3 - 16t - 3)]^2} = \sqrt{1 + (10t + 3)^2} \] Minimize \( D(t) \Rightarrow \) minimize \( (10t + 3)^2 \Rightarrow t = -\frac{3}{10} \) gives minimum value \[ D = \sqrt{1 + 0} = 1 \]