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Question

Accepted Answer

Let us solve the equations step by step:

Given:
1) p cos A = 2q sin A
2) 2p cosec A - q sec A = 3

From (1):
p cos A = 2q sin A
=> p/q = 2 sin A / cos A
=> p/q = 2 tan A
=> p = 2q tan A

Now, substitute p in (2):
2p cosec A - q sec A = 3
Substitute p = 2q tan A:
2(2q tan A) cosec A - q sec A = 3
=> 4q tan A cosec A - q sec A = 3

Recall:
tan A = sin A / cos A
cosec A = 1 / sin A
sec A = 1 / cos A

So:
4q tan A cosec A = 4q (sin A / cos A) * (1 / sin A) = 4q / cos A
q sec A = q / cos A

So equation becomes:
(4q / cos A) - (q / cos A) = 3
=> (3q / cos A) = 3
=> q / cos A = 1
=> cos A = q

Now, from p = 2q tan A:
But tan A = sin A / cos A = sin A / q
So,
p = 2q * (sin A / q) = 2 sin A
=> p = 2 sin A

Recall that cos A = q, so sin^2 A + cos^2 A = 1
=> sin^2 A + q^2 = 1
=> sin^2 A = 1 - q^2
=> sin A = sqrt(1 - q^2)

So, p = 2 sqrt(1 - q^2)

Now, find p^2 + 4q^2:
p^2 = [2 sqrt(1 - q^2)]^2 = 4(1 - q^2) = 4 - 4q^2
So,
p^2 + 4q^2 = (4 - 4q^2) + 4q^2 = 4

Thus, the answer is 4.

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