Question

If a circle passes through $(0,0),(a,0)$ and $(0,b)$, then the coordinates of its centre are

• A. $\left(\frac{b}{2},\frac{a}{2}\right)$
• B. $\left(\frac{a}{2},\frac{b}{2}\right)$
• C. $(b,a)$
• D. $(a,b)$

Answer 1

The correct option is B

$\left(\frac{a}{2},\frac{b}{2}\right)$

Explanation for the correct option:

Step 1. Find the coordinates of centre:

As we know,

The equation of a circle with centre $(h,k)$ and radius as $r$ units, is

${(x–h)}^2+{(y–k)}^2=r^2$

Step 2. Put the values of given coordinates in general equation:

First, $(0,0)$

$\Rightarrow$${(0–h)}^2+{(0–k)}^2=r^2$

$\Rightarrow$ $h^2+k^2=r^2$

Second, $(a,0)$

$\Rightarrow$ ${(a–h)}^2+{(0–k)}^2=r^2$

$\Rightarrow$$a^2+h^2–2ah+k^2=r^2$ …..(1)

Third, $(0,b)$

$\Rightarrow$ ${(0–h)}^2+{(b–k)}^2=r^2$

$\Rightarrow$$h^2+b^2+k^2–2bk=r^2$ …..(2)

Step 3. By solving equation(1) and (2), respectively, we get

$a(a–2h)=0$

$\Rightarrow$ $a=0,2h$

$\Rightarrow$ $h=\frac{a}{2}$

$b(b–2k)=0$

$\Rightarrow$ $b=0,2k$

$\Rightarrow$ $k=\frac{b}{2}$

$\therefore$The coordinates of centre are $\left(\frac{a}{2},\frac{b}{2}\right)$

Hence, Option ‘B’ is Correct.