JEE MAIN - Mathematics (2022 - 27th July Morning Shift - No. 14)
If the circle $$x^{2}+y^{2}-2 g x+6 y-19 c=0, g, c \in \mathbb{R}$$ passes through the point $$(6,1)$$ and its centre lies on the line $$x-2 c y=8$$, then the length of intercept made by the circle on $$x$$-axis is :
$$\sqrt{11}$$
4
3
$$2 \sqrt{23}$$
Explanation
Circle : $${x^2} + {y^2} - 2gx + 6y - 19c = 0$$
It passes through $$h(6,1)$$
$$ \Rightarrow 36 + 1 - 12g + 6 - 19c = 0$$
$$ = 12g + 19c = 43$$ ..... (1)
Line $$x - 2cy = 8$$ passes through centre
$$ \Rightarrow g + 6c = 8$$ ...... (2)
From (1) & (2)
$$g = 2,\,c = 1$$
$$C:{x^2} + {y^2} - 4x + 6y - 19 = 0$$
x intercept $$= 2\sqrt {{g^2} - C} $$
$$ = 2\sqrt {4 + 19} $$
$$ = 2\sqrt {23} $$
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