As observed from the top of the light house of height 'h'm .The angles of depression of two ships approaching it are p and q.If one ship is coming from west and other is coming from south.Find the distance between two ships in terms of a.
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To solve this problem, we can use trigonometry and the concept of angles of depression.
Let's start by visualizing the situation. We have a lighthouse of height 'h' and there are two ships approaching it. One ship is coming from the west, and the other ship is coming from the south.
From the top of the lighthouse, we can draw two lines representing the angles of depression. Let's call these lines AC and BC, where A represents the ship coming from the west, B represents the ship coming from the south, and C represents the top of the lighthouse.
Now, we can form two right-angled triangles: triangle ABC and triangle BDC.
In triangle ABC, the angle of depression is p. This means that the angle formed by line AC and the horizontal line AB is p. Since triangle ABC is a right-angled triangle, the angle between line AC and the vertical line BC is (90 degrees - p).
Similarly, in triangle BDC, the angle of depression is q. This means that the angle formed by line BC and the horizontal line AB is q. Since triangle BDC is a right-angled triangle, the angle between line BC and the vertical line DC is (90 degrees - q).
Now, let's find the distance between the two ships in terms of 'a'.
In triangle ABC, we have the following information:
- The height of the lighthouse is 'h'.
- The angle between line AC and the vertical line BC is (90 degrees - p).
Using trigonometry, we can write the following equation:
tan(90 - p) = h / a
Simplifying the equation, we get:
cot(p) = h / a
Similarly, in triangle BDC, we have the following information:
- The angle between line BC and the vertical line DC is (90 degrees - q).
Using trigonometry, we can write the following equation:
tan(90 - q) = h / a
Simplifying the equation, we get:
cot(q) = h / a
Now, we have two equations with cot(p) and cot(q). By solving these equations simultaneously, we can find the value of 'a' (the distance between the two ships).
Please note that the above explanation assumes that the lighthouse is located at the origin of a coordinate plane, and the ships are approaching along the positive x-axis (west) and positive y-axis (south). If the lighthouse is located at a different position or the ships are approaching from different directions, the approach may vary.