Question

The points of contact of the vertical tangents $x=2-3\sin \theta$, $y=3+2\cos \theta$ are

• A. $(2,5),(2,1)$
• B. $(-1,3),(5,3)$
• C. $(2,5),(5,3)$
• D. $(-1,3),(2,1)$

Answer 1

Answer: (B)

$(-1,3),(5,3)$

For the tangents to be vertical,
$\frac{dy}{dx}=\infty$
Or
$\frac{dx}{dy}=0$
Or
$\frac{dx}{d\theta }=0$
Or
$-3\cos \theta =0$
Or
$\theta =\frac{2n-1}{2}\pi$
Hence
$\theta =\frac{\pi }{2},\frac{3\pi }{2}$.
Now
$x_{\frac{\pi }{2}}$
$=-1$
$y_{\frac{\pi }{2}}$
$=3$.
Similarly
$x_{\frac{3\pi }{2}}=5$
$y_{\frac{3\pi }{2}}=3$.
Hence the points are
$(-1,3)$ and $(3,5)$.