Question

Circle(s) touching $X$-axis at a distance $3$ from the origin and having an intercept of length $2\sqrt{7}$ on $Y$-axis is/are

• A. $x^2+y^2-6x+8y+9=0$
• B. $x^2+y^2-6x+7y+9=0$
• C. $x^2+y^2-6x-8y+9=0$
• D. $x^2+y^2-6x-7y+9=0$

Answer 1

Answer: (A), (C)

PLAN

Here, the length of intercept on $Y$-axis is
$\Rightarrow 2\sqrt{f^2}-c$ and if circle touches $X$-axis
$\Rightarrow g^2=c$
for $x^2+y^2+2gx+2fy+c=0$
Here, $x^2+y^2+2gx+2fy+c=0$

passes through $(3,0)$.
$\Rightarrow 9+6g+c=\mathrm{0...}(i)$
$g^2=c...(ii)$
and $2\sqrt{f^2-c}=2\sqrt{7}$
$f^2-c=\mathrm{7...}(iii)$
From Eqs. (i) and (ii), we get
$g^2+6g+9=0$
$\Rightarrow (g+3{)}^2=0$
$\Rightarrow (g=-3)$ and $c=9$
$\therefore f^2=16\Rightarrow f=\pm 4$
$\therefore x^2+y^2-6x\pm 8y+9=0$