Case Study 1: Read the following passage and answer the questions given below.
The Indian coast guard, while patrolling, saw a suspicious boat with people. They were nowhere looking like fishermen. The coast guard was closely observing the movement of the boat for an opportunity to seize the boat. They observed that the boat is moving along a planar surface. At an instant of time, the coordinates of the position of the coast guard helicopter and the boat are (1, 3, 5) and (2, 5, 3) respectively.

Answer the following questions using the above information.
(i) If the line joining the positions of the helicopter and the boat is perpendicular to the plane in which the boat moves, then find the equation of the plane.
(ii) If the coast guard decide to shoot the boat at that given instant of time, then what is the distance (in metres) that the bullet has to travel?
(iii) If the coast guard decides to shoot the boat at that given instant of time, when the speed of bullet is 36m/s, then what is the time taken for the bullet to travel and hit the boat?

Question

Case Study 1: Read the following passage and answer the questions given below.
The Indian coast guard, while patrolling, saw a suspicious boat with people. They were nowhere looking like fishermen. The coast guard was closely observing the movement of the boat for an opportunity to seize the boat. They observed that the boat is moving along a planar surface. At an instant of time, the coordinates of the position of the coast guard helicopter and the boat are (1, 3, 5) and (2, 5, 3) respectively.

Answer the following questions using the above information.
(i) If the line joining the positions of the helicopter and the boat is perpendicular to the plane in which the boat moves, then find the equation of the plane.
(ii) If the coast guard decide to shoot the boat at that given instant of time, then what is the distance (in metres) that the bullet has to travel?
(iii) If the coast guard decides to shoot the boat at that given instant of time, when the speed of bullet is 36m/s, then what is the time taken for the bullet to travel and hit the boat?

Accepted Answer

DR's of normal to the plane line joining points
$(2, 5, 3) and (1, 3, 5) i.e. (1 - 2, 3 - 5, 5 - 3) = (- 1, - 2, 2) . Equation of plane passing through (2, 5, 3) and having direction ratios (- 1, - 2, 2)$
$\begin{array}{l} (- 1) (x - 2) + (- 2) (y - 5) + (2) (z - 3) = 0 \\ \Rightarrow - x + 2 - 2 y + 10 + 2 z - 6 = 0 \\ \Rightarrow - x - 2 y + 2 z + 12 - 6 = 0 \\ \Rightarrow x + 2 y - 2 z = 6 \end{array}$
$(ii) Let A be (1, 3, 5) and B be (2, 5, 3) . Then, AB = \sqrt{(2 - 1)^{2} + (5 - 3)^{2} + (3 - 5)^{2}} \begin{array}{l} = \sqrt{(1)^{2} + (2)^{2} + (- 2)^{2}} \\ = \sqrt{1 + 4 + 4} = \sqrt{9} = 3 m \end{array}$
$(iii) ∴ Required Time = \frac{3}{36} = \frac{1}{12} \sec$

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