Question

Circle(s) touching $X$-axis at a distance $3$ from the origin and having an intercept of length $2 \sqrt7$ 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 $(3,0)$.
$\Rightarrow 9 + 6 g + c = 0 ...(i)$
$ g^2 = c ...(ii)$
and $ 2 \sqrt {f^2- c} = 2 \sqrt7 $
$f^2 - c = 7 ...(iii)$
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 $