Question

If the circle $x^2+y^2-2gx+6y-19c=0,g$, $c\in R$ passes through the point $(6,1)$ and its centre lies on the line $x-2cy=8$, then the length of intercept made by the circle on $x$-axis is

• A. $\sqrt{11}$
• B. 4
• C. 3
• D. $2\sqrt{23}$

Answer 1

Answer: (D)

Given circle $x^2+y^2-2gx+6y-19c=0$
Passes through $(6,1)$
$12g+19c=\mathrm{43.....}$(1)
Centre $(g,-3)$ lies on given line
So, $g+6c=\mathrm{8.....}$(2)
Solve equation (1) \& (2)
$c=1\&g=2$
equation of circle $x^2+y^2-4x+6y-19=0$
Length of intercept on $x$-axis
$=2\sqrt{g^2-c}=2\sqrt{23}$