Let the circle C_1:x^2+y^2-2(x+y)+1=0 and C_2 be a circle having centre at (-1,0) and radi…

Question

Let the circle ${C}_{1}:{x}^{2}+{y}^{2}-2(x+y)+1=0$ and ${C}_{2}$ be a circle having centre at $(-1,0)$ and radius 2 . If the line of the common chord of ${C}_{1}$ and ${C}_{2}$ intersects the $y$-axis at the point $P$, then the square of the distance of P from the centre of ${C}_{1}$ is:

Accepted Answer

$\begin{gathered} {C}_{1}:{x}^{2}+{y}^{2}-2(x+y)+1=0 \ {C}_{2}:(x+1{)}^{2}+{y}^{2}=(2{)}^{2} \ {x}^{2}+{y}^{2}+2x-3=0 \end{gathered}$

Common chord is

$\begin{gathered} {C}_{1}-{C}_{2}=0 \ \Rightarrow 2x+y-2=0 \end{gathered}$

also, this line intersects the $y$-axis at the point

$\begin{gathered} P(y,0). \ \Rightarrow y=2 \end{gathered}$

$P(2,0)$

Distance of point $P$ from $(1,1)$ is

$\begin{gathered} d=\sqrt{(2-1{)}^{2}+(0-1{)}^{2}} \ =\sqrt{{1}^{2}+{1}^{2}} \ d=\sqrt{2} \ \Rightarrow {d}^{2}=2 \end{gathered}$

Solutions · 1