To find the shortest distance from the surface xy+6x+z^2=33 to the origin, we need to find the perpendicular distance from the origin to the surface.
The perpendicular distance from a point (x0, y0, z0) to a plane Ax + By + Cz + D = 0 is given by the formula:
d = |Ax0 + By0 + Cz0 + D| / sqrt(A^2 + B^2 + C^2)
In this case, the equation of the surface can be written as:
xy + 6x + z^2 - 33 = 0.
Comparing this to the general equation of a plane Ax + By + Cz + D = 0, we have:
A = 1, B = 0, C = 1, D = -33.
The coordinates of the origin are (0, 0, 0).
Using the formula for perpendicular distance, we can calculate:
d = |1(0) + 0(0) + 1(0) + (-33)| / sqrt(1^2 + 0^2 + 1^2)
= 33 / sqrt(2)
Therefore, the shortest distance from the surface xy+6x+z^2=33 to the origin is 33 / sqrt(2).
What is the shortest distance from the surface xy+6x+z2=33
to the origin?
1 answer