What is the shortest distance from the surface xy+6x+z2=33
to the origin?
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).
To find the shortest distance from the surface to the origin, we need to find the perpendicular distance between the origin and the surface.
Step 1: Rewrite the equation of the surface in the form of Ax + By + Cz + D = 0.
Given equation: xy + 6x + z^2 = 33
To rewrite it in the desired form, we need to complete the square for the quadratic term.
Let's complete the square for z^2:
(xy + 6x) + (z^2) = 33
(xy + 6x) + (z^2 + 12z + 36) = 33 + 36
(xy + 6x + 36) + (z^2 + 12z + 36) = 69
Simplifying, we have:
(xy + 6x + 36) + (z^2 + 12z + 36) - 69 = 0
xy + 6x + z^2 + 12z + 36 - 69 = 0
xy + 6x + z^2 + 12z - 33 = 0
So, we have the surface equation in the desired form: Ax + By + Cz + D = 0.
A = 1, B = 0, C = 1, D = -33.
Step 2: Calculate the distance between the origin and the surface using the formula:
distance = |A*0 + B*0 + C*0 + D| / sqrt(A^2 + B^2 + C^2)
In this case, A = 1, B = 0, C = 1, D = -33.
distance = |0 + 0 + 0 - 33| / sqrt(1^2 + 0^2 + 1^2)
distance = |-33| / sqrt(1 + 0 + 1)
distance = 33 / sqrt(2)
Therefore, the shortest distance from the surface xy + 6x + z^2 = 33 to the origin is 33 / sqrt(2).