Answer:
(x - 5)^2 + (y - 5)^2 = 25
Explanation:
If a circle has no x-intercepts and no y-intercepts, it means that the circle does not intersect either the x-axis or the y-axis. In other words, the center of the circle must lie at some point (h, k) where h and k are not equal to zero.
The general equation of a circle with center (h, k) and radius r is:
(x - h)^2 + (y - k)^2 = r^2
Since we know that the circle has a radius of 5, we can substitute r = 5 into the equation:
(x - h)^2 + (y - k)^2 = 25
Now, we need to find values for h and k such that the circle has no x-intercepts and no y-intercepts. Since the x-intercepts occur when y = 0 and the y-intercepts occur when x = 0, we need to make sure that neither of these values satisfies the equation.
Let's first consider the x-intercepts. We want to make sure that the circle does not intersect the x-axis, which means that the y-coordinate of the center must be equal to the radius. In other words, k = 5. Now, the equation becomes:
(x - h)^2 + (y - 5)^2 = 25
Next, we need to make sure that the circle does not intersect the y-axis, which means that the x-coordinate of the center must be equal to the radius. In other words, h = 5. Now, the equation becomes:
(x - 5)^2 + (y - 5)^2 = 25
Therefore, the equation of the circle that has no x-intercepts, no y-intercepts, and a radius of 5 is:
(x - 5)^2 + (y - 5)^2 = 25