$ ext{Find all circle equations that lie on the unit circle}$

The unit circle is defined by the equation:

$x^2 + y^2 = 1$

To find all circles that lie on the unit circle, we consider the general equation of a circle:

$ (x – a)^2 + (y – b)^2 = r^2 $

For the circle to lie on the unit circle, the radius of this circle must be zero, as any larger radius would extend beyond the unit circle. Therefore:

$ r = 0 $

Thus, the equation simplifies to a point:

$ (x – a)^2 + (y – b)^2 = 0 $

Expanding this gives:

$ x = a, y = b $

But since it must lie on the unit circle:

$ a^2 + b^2 = 1 $

So, all such points (a, b) lie on the unit circle.

Therefore, the equations of all circles on the unit circle are:

$ (x – a)^2 + (y – b)^2 = 0 $ where $ a^2 + b^2 = 1 $

The unit circle has the equation:

$ x^2 + y^2 = 1 $

The equation for any circle is:

$ (x – h)^2 + (y – k)^2 = r^2 $

For a circle to be on the unit circle, its center (h, k) must be on the unit circle. Thus:

$ h^2 + k^2 = 1 $

The radius of the circle must also be zero because any radius larger than zero would extend beyond the unit circle. Therefore:

$ r = 0 $

This simplifies the circle’s equation to:

$ (x – h)^2 + (y – k)^2 = 0 $

which is equivalent to:

$ x = h, y = k $

Thus, the center (h, k) is a point on the unit circle, giving us:

$ h^2 + k^2 = 1 $

Therefore, the equations of all such circles are:

$ (x – h)^2 + (y – k)^2 = 0 $ where $ h^2 + k^2 = 1 $

A unit circle satisfies:

$ x^2 + y^2 = 1 $

The general circle equation is:

$ (x – h)^2 + (y – k)^2 = r^2 $

For a circle to lie on the unit circle:

$ h^2 + k^2 = 1, r = 0 $

Thus, the equations are:

$ (x – h)^2 + (y – k)^2 = 0 $ where $ h^2 + k^2 = 1 $

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