The length of the chord intercepted by the ci | vedclass

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(C) The equation of the circle is $x^2+y^2-4x+6y-12=0$. Comparing this with $x^2+y^2+2gx+2fy+c=0$,we get $g=-2, f=3, c=-12$. The center of the circle

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$10$

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(C) The equation of the circle is $x^2+y^2-4x+6y-12=0$. Comparing this with $x^2+y^2+2gx+2fy+c=0$,we get $g=-2, f=3, c=-12$. The center of the circle is $(-g, -f) = (2, -3)$ and the radius $r = \sqrt{g^2+f^2-c} = \sqrt{(-2)^2+3^2-(-12)} = \sqrt{4+9+12} = \sqrt{25} = 5$. The perpendicular distance $d$ from the center $(2, -3)$ to the line $4x+3y+1=0$ is given by $d = \frac{|4(2)+3(-3)+1|}{\sqrt{4^2+3^2}} = \frac{|8-9+1|}{\sqrt{16+9}} = \frac{0}{5} = 0$. Since the perpendicular distance $d=0$,the line passes through the center of the circle,meaning the chord is a diameter. The length of the chord is $2 	imes \sqrt{r^2-d^2} = 2 	imes \sqrt{5^2-0^2} = 2 	imes 5 = 10$.

The length of the chord intercepted by the circle $x^2+y^2-4x+6y-12=0$ on the line $4x+3y+1=0$ is

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