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)

The correct options are
A $x^2+y^2-6x+8y+9=0$
C $x^2+y^2-6x-8y+9=0$

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