Question

If $x+y=\cos \theta ,x-y=\sin \theta \left(\theta \text{is parameter}\right)$, then the locus of $P(x,y)$ is

• A. a straight line of slope $\frac{1}{2}$
• B. a circle with centre $(0,0)$
• C. a straight line passing through $(0,0)$
• D. a circle with radius $\frac{1}{\sqrt{2}}\text{units}$

Answer 1

Answer: (B), (D)

The correct options are
B a circle with centre $(0,0)$
D a circle with radius $\frac{1}{\sqrt{2}}\text{units}$

Given equations,
$x+y=\cos \theta \dots \left(1\right)x-y=\sin \theta \dots \left(2\right)$
From $(1{)}^2+(2{)}^2$
$2(x^2+y^2)=1\Rightarrow x^2+y^2=\frac{1}{2}$
$\Rightarrow x^2+y^2={\left(\frac{1}{\sqrt{2}}\right)}^2$
Hence, locus of $P(x,y)$ is a circle with centre at $(0,0)$ and whose radius is $\frac{1}{\sqrt{2}}\text{units}$