Question

Consider a circle with parametric equation $x=\frac{a}{\sqrt{2}}+a\cos \theta ,$ $y=\frac{a}{\sqrt{2}}+a\sin \theta .$ Which of the following is/are TRUE ?

• A. circle passes through the origin
• B. radius of the circle is $\frac{a}{\sqrt{2}}$ units
• C. circle passes through $(\frac{a}{2},\frac{a}{2})$
• D. radius of the circle is $a$ units

Answer 1

Answer: (A), (D)

radius of the circle is $a$ units

Given $x=\frac{a}{\sqrt{2}}+a\cos \theta ,y=\frac{a}{\sqrt{2}}+a\sin \theta$
$\Rightarrow x-\frac{a}{\sqrt{2}}=a\cos \theta \dots \left(1\right)$
and $y-\frac{a}{\sqrt{2}}=a\sin \theta \dots \left(2\right)$
Squaring and adding equations $\left(1\right)$ and $\left(2\right)$
${(x-\frac{a}{\sqrt{2}})}^2+{(y-\frac{a}{\sqrt{2}})}^2=a^2{\cos }^2\theta +a^2{\sin }^2\theta$
$\Rightarrow {(x-\frac{a}{\sqrt{2}})}^2+{(y-\frac{a}{\sqrt{2}})}^2=a^2$

Check for $(0,0)$
${(0-\frac{a}{\sqrt{2}})}^2+{(0-\frac{a}{\sqrt{2}})}^2=a^2$
So, circle passes through the origin.

Check for $(\frac{a}{2},\frac{a}{2})$
${(\frac{a}{2}-\frac{a}{\sqrt{2}})}^2+{(\frac{a}{2}-\frac{a}{\sqrt{2}})}^2=a^2+\frac{a^2}{2}-\sqrt{2}{a}^2\ne a^2$
So, circle does not pass through $(\frac{a}{2},\frac{a}{2})$