Diametrically-opposite means the two points are on opposite ends of a diameter. Meaning, the line connecting the two points is a diameter of the circle.
A diameter passes through the center of the circle. A diameter bisects a circle, so the two sectors formed by a diameter have 180 degrees central angle each.
Since the meaure of the arc subtended by a central angle is equal to the measure of the central angle, then a diameter bisects the circumference into two 180-degree arcs.
The measure of an inscribed angle, meaning its vertex is on the circumference, is half of the measure of the circular arc subtended by the angle.
So, the measure of an inscribed angle whose endpoints are diametrically opposite is (1/2)(180deg) = 90 degrees. --------------(i)
In your Problem above, it should have been "points A and B are diametrically opposite to the point X", not the posted point A, so that points A,Y,B have a chance to be collinear.
Draw the figure on paper. The two circles need not be equal , or the two circles may be different in sizes.
Draw the points X,Y,A,B.
Draw the line segment XY.
In triangle XYB,
angle XYB = 90 degrees
In triangle XYA,
angle XYA = 90 degrees
Hence, angle AYB = 90 +90 = 180 degrees.
So, AYB is a straight line.
Therefore, A,Y,B are collinear. ----------------proven.