Q: What equation represents the family of circles through the origin and with center on the x-axis?

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x2 + y2 = R2

Semi-circles

concentric circles

Concentric circles are the circles with the same center therefore they do not cross with each other as the "center is not considered a point on the circle". An exception would be two circles that are concentric and have the same radius, in which case the circles are indistinct and every point of the circles is an intersection.

a diameter

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The arrow, which represents "produces" or "yields".

By using Cartesian equations for circles on the Cartesian plane

x2 + y2 = R2

All parallels of latitude, except for the Equator, are not great circles. Great circles are the largest circles that can be drawn on a sphere and pass through its center, whereas small circles do not pass through the center of the sphere.

Circles in the same plane with the same center are concentric circles.

Semi-circles

If the tangent circles are outside of one another, then neither passes through the center of the other. If one circle is within the other, then the inner tangent circle might contain the center point of the larger circle. There will be infinitely many inner tangent circles that do not.

Yes, it is. Circles that are in the same plane and having the same center are called congruent circles.

concentric circles

concentric circles

Concentric circles are the circles with the same center therefore they do not cross with each other as the "center is not considered a point on the circle". An exception would be two circles that are concentric and have the same radius, in which case the circles are indistinct and every point of the circles is an intersection.