Question

If a circle passes through the point $(0,0),(a,0),(0,b)$,then its centre is

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

Answer 1

The correct option is C

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

Explanation for the correct option:

Step 1. Find the coordinates of centre:

As we know,

the equation of circle is $x^2+y^2+2gx+2fy$ with centre $\left(-g,-f\right)$ and radius $r$

Step 2. On passing through the points $(0,0),(a,0),(0,b)$, we get

At, $(0,0)$

$c=0$

At, $(a,0)$

$a^2+2ga+c=0$

$\Rightarrow$ $2ga=-a^2$

$\Rightarrow$ $g=-\frac{a}{2}$

At, $(0,b)$

$b^2+2fb+c=0$

$\Rightarrow$ $2fb=-b^2$

$\Rightarrow$ $f=-\frac{b}{2}$

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

Hence, Option ‘C’ is Correct.