Let a circle $C$ pass through the points $(4,2)$ and $(0,2)$, and its centre lie on $3 x+2 y+2=0$. Then the length of the chord, of the circle $C$, whose mid-point is $(1,2)$, is :

Question

Aicharya · Accepted Answer

Let the centre be
$\left(-2 a, \frac{6 a-2}{2}\right) \equiv(-2 a, 3 a-1)$
Centre is equal distance from $(4,2)$ and $(0,2)$
$\begin{aligned} & \Rightarrow \sqrt{(4+2 a)^2+(3 a-3)^2}=\sqrt{(-2 a-0)^2+(3 a-3)^2} \\ & \Rightarrow(2 a+4)^2+9(a-1)^2=4 a^2+9(a-1)^2 \\ & \Rightarrow 4 a^2+16+16 a=4 a^2 \Rightarrow a=-1 \\ & \Rightarrow \text { centre } \equiv(2,-4) \Rightarrow \text { Radius }=\sqrt{40}\end{aligned}$

$\begin{aligned} & \Rightarrow A M^2=(\sqrt{40})^2-(\sqrt{37})^2 \\ & \Rightarrow 2 A M=A B=2 \sqrt{3}\end{aligned}$

Let a circle $C$ pass through the points $(4,2)$ and $(0,2)$, and its centre lie on $3 x+2 y+2=0$. Then the…

Solution