Question

Find the equation of the sphere center at \((2,-3,4)\) and tangent to the \(xz\) plane.

Short answer

The equation of sphere is \(x^2+y^2+z^2-4x+6y-8z+20=0\).

Step-by-step solution

Given Information

The center at \((2,-3,4)\).

The tangent plane to the sphere is (\xz\) plane.

Find the radius

Since \(xz\) plane is tangent of the sphere, the radius is the y-coordinate of the center.

Therefore, the radius is \(r=3\).

Find the equation of sphere

Substitute \((2,-3,4)\) and \(r=3\) into equation of sphere.

\((x-2)^2+(y+3)^2+(z-4)^2=3^2\).

Simplify the equation and the equation is,

\(x^2+y^2+z^2-4x+6y-8z+20=0\).