Two circles with equal radii are intersecting at the points (0,1) and (0,-1). The tangent at the point (0,1) to one of the circles is given. Then the distance between the centres of these circles is:

The correct option is D

Let the two circles with equal radii intersect at points A(0,1) and B(0,-1). Let the centers of the circles be C1 and C2. Let the tangent at point A(0,1) to one of the circles be drawn. The distance between the centers of the circles is the distance between C1 and C2.

Consider the triangle formed by connecting the centers C1 and C2 and the point of intersection A(0,1). This triangle is an isosceles triangle because the radii are equal. Let's call the radius r. The distance between A and either center is r. The distance between A and B is 2. The line connecting A and B is perpendicular to the line segment connecting the centers C1 and C2. Therefore, we have two right-angled triangles, each with hypotenuse r and one leg of length 1 (half the distance between A and B).

By the Pythagorean theorem, we have:

r² = 1² + (d/2)²
where d is the distance between the centers C1 and C2.

Since the circles have equal radii, the distance between the centers is twice the distance from the center of one circle to the point (0,1), that is 2*1=2.

Therefore, the distance between the centres of these circles is 2.