Q115 / 180 · KCET 2016 · Mathematics
If \( x=2+3 \cos \theta \) and \( y=1-3 \sin \theta \) represent a circle then the centre and radius is
- A\( (2,1), 9 \)
- Correct answerB\( (2,1), 3 \)
- C\( (1,2), \frac{1}{3} \)
- D\( (-2,-1), 3 \)
Solution
Given that,
$x=2+3 \cos \theta \rightarrow(1)$
$y=1-3 \sin \theta \rightarrow(2)$
From Eqs. $(1)$ and $(2)$, we have
$\frac{x-2}{3}=\cos \theta$ and $\frac{y-1}{-3}=\sin \theta$
We know that, $\sin ^{2} \theta+\cos ^{2} \theta=1$
So, $\left(\frac{x-2}{3}\right)^{2}+\left(\frac{y-1}{-3}\right)=1$
$\Rightarrow \frac{(x-2)^{2}}{9}+\frac{(y-1)^{2}}{9}=1$
$\Rightarrow(x-2)^{2}+(y-1)^{2}=9$
So, it is circle centre at point $(2,1)$ and radius is 3 .
Hence, the correct answer is option: 'B'.