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Question: Find the points of horizontal tangency to the polar curve r = 5 cosecθ + 1.


A ) (6, π2) and (4, 3π2)
B ) (- 6, π2) and (4, 3π2)
C ) (- 6, π2) and (- 4, 3π2)
D ) (6, π2) and (6, 3π2)
E ) (6, π2) and (- 4, 3π2)

Steps to derive

1 r = 5 cosecθ + 1
 [Write the polar equation.]

2 Points of horizontal tangency are obtained by solving dydθ = 0

3 y = r sinθ = sinθ (5 cosecθ + 1) = 5 + sinθ
 [r = 5 cosecθ + 1.]

4 dydθ = cosθ

5 dydθ = 0, so cosθ = 0, θ = π2, 3π2

6 At θ = π2, r = 6 and at θ = 3π2, r = - 4

7 So, (6, π2) and (- 4, 3π2) are the points of horizontal tangency to the polar curve.



Hence the right answer is Option E

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