# Two circles with equal radii intersecting at the points (0,1) and (0,-1). The tangent at the point (0,1) to one of the circles passes through the centre of the other circle.Then the distance between the centres of these circles is:Option 1)  2Option 2)  $2\sqrt2$Option 3)  $\sqrt2$Option 4)  1

Orthogonality of two circle -

Two circles $S_{1}=0$ and $S_{2}=0$  are said to be orthogonal ,if tangents at their point of intersection include right angle.

- wherein

$2g_{1}g_{2}+2f_{1}f_{2}=c_{1}c_{2}$

Let centre of circle are $\left ( c,0 \right )$ &  $\left ( -c,0 \right )$

$\Rightarrow$ equation of circles are

$\left ( x-1 \right )^{2}+\left ( y-0 \right )^{2}=c^{2}+1$

and

$\left ( x+1 \right )^{2}+\left ( y-0 \right )^{2}=c^{2}+1$

$\Rightarrow$ since circles are orthogonal so ,

$2\left ( c \right )\left ( -c \right )+\left ( 0 \right )\left ( 0 \right )=\left ( -1 \right )+\left ( -1 \right )$

$\Rightarrow c^{2}=1\Rightarrow c=\pm 1$

Distance between $\left ( -1,0 \right )$ &  $\left ( 1,0 \right )$ is $2$

Option 1)

2

Option 2)

$2\sqrt2$

Option 3)

$\sqrt2$

Option 4)

1

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