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 \(\mathrm{C}_{1}\) and \(\mathrm{C}_{2}\) intersects the y-axis at the point \(\mathrm{P}\), then the square of the distance of \(\mathrm{P}\) from the centre of \(\mathrm{C}_{1}\) is :

(1) 2

(2) 1

(3) 6

(4) 4

Answer 1

Correct option is (1) 2

\(S_{1}: x^{2}+y^{2}-2 x-2 y+1=0\)

\(S_{2}: x^{2}+y^{2}+2 x-3=0\)

Common chord \(=\mathrm{S}_{1}-\mathrm{S}_{2}=0\)

\(-4 \mathrm{x}-2 \mathrm{y}+4=0\)

\(2 \mathrm{x}+\mathrm{y}=2 \Rightarrow \mathrm{P}(0,2)\)

\(\mathrm{d}_{(\mathrm{c}, \mathrm{p})}^{2}=(1-0)^{2}+(2-1)^{2}=2\)