First slide

Tangents and normals

Question

The points of contact of the vertical tangents to $x = 2 - 3 \sin θ, y = 3 + 2 \cos θ a r e$

Moderate

A

(2, 5), (2, 1)
B

(-1 , 3), (5, 3)
C

(2, 5), (5, 3)
D

(-1, 3), (2, 1)

Solution

$\begin{array}{l} For vertical tangents \frac{d x}{d θ} = 0 so, we have \\ - 3 \cos θ = 0 \Rightarrow θ = \frac{π}{2} or \frac{3 π}{2} . Corresponding to these \end{array}$
$\begin{array}{l} values of θ, we have \\ x = 2 - 3 \sin \frac{π}{2} = - 1, y = 3 + 2 \cos \frac{π}{2} = 3 \\ x = 2 - 3 \sin \frac{3 π}{2} = 2 + 3 = 5, y = 3 + 2 \cos \frac{3 π}{2} = 3 \end{array}$
Thus the required points are (-1, 3) , (5, 3).

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