Question

Two circles with equal radii areintersecting at the points (0, 1) and (0,-1). The tangent at the point (0.1) to oneof the circles passes through the centreof the other circle.Then the distancebetween the centres of these circle is _____.

Answer 1

The distance between the centres of the two intersecting circles with a common radius is twice the radius of the circles.

Step-by-step explanation:

To solve for the distance between the centres of the two intersecting circles, let's denote their centres as C1 and C2 and their common radius as r. Given that the circles intersect at points (0,1) and (0,-1), the line connecting these points is a vertical line, which is perpendicular to the line containing the centres C1 and C2. This makes the line segment C1C2 a horizontal line.

The tangent at the point (0,1) to one of the circles will be horizontal, and since it passes through the centre of the other circle, its equation will be y=1. Therefore, the centre of this circle must lie on the line y=1 and at a distance r from (0,1), which is a point on the circle. The centre of the other circle must then be located on the same horizontal line but on the opposite side of the vertical line.

Since the distance from (0,1) or (0,-1) to each centre is r, and these points lie on the circle itself, the total distance between C1 and C2 will be 2r. Hence, the distance between the centres of these circles is twice the radius.