Circle 1 Question 12
12. 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) $x^{2}+y^{2}-6 x+8 y+9=0$
(2013 Adv.)
(b) $x^{2}+y^{2}-6 x+7 y+9=0$
(c) $x^{2}+y^{2}-6 x-8 y+9=0$
(d) $x^{2}+y^{2}-6 x-7 y+9=0$
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Answer:
Correct Answer: 12. (c)
Solution:
- PLAN
Here, the length of intercept on $Y$-axis is $\Rightarrow 2 \sqrt{f^{2}-c}$ and if circle touches $X$-axis
passes through $(3,0)$.
$$ \begin{aligned} & \Rightarrow \quad 9+6 g+c=0 \\ & \begin{aligned} g^{2} & =c \\ 2 \sqrt{f^{2}-c} & =2 \sqrt{7} \end{aligned} \\ & f^{2}-c=7 \end{aligned} $$
From Eqs. (i) and (ii), we get
$$ \begin{aligned} & & g^{2}+6 g+9 & =0 \Rightarrow(g+3)^{2}=0 \\ & \Rightarrow & g & =-3 \text { and } c=9 \\ & \therefore & f^{2} & =16 \Rightarrow f= \pm 4 \\ & \therefore & x^{2}+y^{2}-6 x \pm 8 y+9 & =0 \end{aligned} $$
