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If the circle $\mathrm{x^{2}+y^{2}-2 g x+6 y-19 c=0, g, c \in \mathbb{R}}$ passes through the point $\mathrm{(6,1)}$ and its centre lies on the line $\mathrm{ x-2 c}$ $\mathrm{ y=8,}$ then the length of intercept made by the circle on $\mathrm{ x-axis}$ is
Option: 1
$\sqrt{11}$

Option: 2
$4$

Option: 3
$3$

Option: 4
$2 \sqrt{23}$

Answers (1)

best_answer

$\text{Given circle} \; \; \mathrm{x^{2}+y^{2}-29 x+6 y-19 c=0}$

$\text{passes through} (6,1)$

$\mathrm{12 g+19 c=43} \cdots \text { (1) }$

Centre (g , -3) lies on given line

$\text{So}, \;\; \mathrm{g+6 c=8 } \cdots (2)\\$

Solve equation (1) & (2)

$c=1 \;\; \& \;\; y=2 \\$

$\text{equation of circle}\;\; \mathrm{ x^{2}+y^{2}-4 x+6 y-19=0}$

Length of intercept on x-axis

$=2\sqrt{\mathrm{g}^{2}-\mathrm{c}}$ $=2\sqrt{23}$

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