Question

Centre of the circle touching $y-$axis at $(0,3)$ and making an intercept $2$ units on positive $X-$axis is

• A. $(\sqrt{10},3)$
• B. $(3\sqrt{10},)$
• C. $(-\sqrt{10},3)$
• D. $(-\sqrt{10},-3)$

Answer 1

The correct option is A $(\sqrt{10},3)$

Let the eq. of the circle is

$x^2+y^2+2gx+2fy+c=\mathrm{0.....}\left(1\right)$

eq $\left(1\right)$ touches $y-$axis

$c=f^2.....\left(2\right)$

Point $(0,3)$ lies on circle

$x=0,y=3$ in $\left(1\right)$

$9+6f+f^2=0$

$f^2+\mathrm{2.3.}f+3^2=0$

$(f+3{)}^2=0$

$f+3=0\to f=-3$

$c=f^2=(-3{)}^2=9$

Circle cuts an intercept $2$ units on $+ve{}^{\prime }{x}^{\prime }axis$

$2\sqrt{g^2-c}=2$

$\sqrt{g^2-9}=1$

$g^2-9=1$

$g^2=10$

$g=\sqrt{10}$

Centre $=(\sqrt{10},3)$