a^2+b^2+c^2=/0,what's the equation of lines: (x-1)/a=(y-1)/b=(z-1)/c rotate around z axis

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We have a line in 3D space with equation (x-1)/a equals (y-1)/b equals (z-1)/c, where a squared plus b squared plus c squared is not zero. We need to find the equation of the surface formed when this line rotates around the z-axis. The line passes through point (1,1,1) and has direction vector (a,b,c). To find the surface equation, we first parameterize the line. Any point on the line can be written as x equals 1 plus a t, y equals 1 plus b t, z equals 1 plus c t, where t is a parameter. When this line rotates around the z-axis, each point traces a circle. The rotation conditions are: the z-coordinate remains the same, and the distance from the z-axis is preserved. Now we eliminate the parameter t. From z equals 1 plus c t, we solve for t equals z minus 1 over c, assuming c is not zero. We substitute this expression for t into the distance equation. This gives us x squared plus y squared equals the sum of two squared terms involving z. We simplify the equation by multiplying both sides by c squared to clear the denominators. This gives us the final surface equation: c squared times the sum of x squared plus y squared equals the sum of two squared terms. This is the equation of the surface of revolution formed when the original line rotates around the z-axis. In conclusion, the equation of the surface formed by rotating the line around the z-axis is c squared times x squared plus y squared equals the sum of two squared terms involving z. When c equals zero, the surface degenerates to the plane z equals 1. This quadratic surface represents all points generated by the rotation of the original line around the z-axis.