Intersection of a circle with coordinate planes

• Calpalned
In summary, an equation for a sphere with center (2, -6, 4) and radius 5 is (x - 2)^2 + (y + 6)^2 + (z - 4)^2 = 25. This sphere intersects the XY plane with a circle of radius 3 and the YZ plane with a circle of radius (21)^0.5. However, the XZ plane does not intersect with the sphere as the radius squared is less than zero. This is due to one of the starting points being farther from the axis than the radius of the sphere.
Calpalned

Homework Statement

Find an equation of the sphere with center (2, -6, 4) and radius 5. Describe its intersection with the each of the coordinate planes.

Homework Equations

Equation of a sphere with three dimensions X2 + Y2 + Z2 = R2

The Attempt at a Solution

My equation is (x - 2)2 + (y + 6)2 + (z - 4)2 = 25
This sphere intersects the XY plane by making a circle with radius 3 and it intersects the YZ plane with a radius of (21)0.5. However, my equation for the XZ plane came out to be negative. That is, Radius squared is less than zero. Does that mean the sphere doesn't intersect the XZ plane?

Since one of your starting points (-6) is farther from an axis than the radius of the sphere (5), it seems reasonable that the sphere wouldn't intersect with one of the planes.

Borg said:
Since one of your starting points (-6) is farther from an axis than the radius of the sphere (5), it seems reasonable that the sphere wouldn't intersect with one of the planes.
Thanks Borg. I never noticed the correlation between the points and the radius before.

1. What is the equation for the intersection of a circle with the x-axis?

The equation for the intersection of a circle with the x-axis is x2 + y2 = r2, where r is the radius of the circle.

2. How do you find the x-intercepts of a circle?

To find the x-intercepts of a circle, set y = 0 in the equation x2 + y2 = r2 and solve for x. The resulting values of x will be the x-intercepts of the circle.

3. What is the equation for the intersection of a circle with the y-axis?

The equation for the intersection of a circle with the y-axis is x2 + y2 = r2, where r is the radius of the circle.

4. How do you find the y-intercepts of a circle?

To find the y-intercepts of a circle, set x = 0 in the equation x2 + y2 = r2 and solve for y. The resulting values of y will be the y-intercepts of the circle.

5. Can a circle intersect with the xy-plane?

No, a circle cannot intersect with the xy-plane as the xy-plane is a two-dimensional plane and a circle is a three-dimensional object. However, the projection of a circle onto the xy-plane will result in an ellipse.

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