To find the shortest distance from the surface to the origin, we can use the formula for the distance between a point and a plane.
The given equation of the surface can be rewritten as:
z^2 = 33 - xy - 6x
We can treat this equation as the equation of a function f(x,y) = 33 - xy - 6x.
To find the shortest distance, we need to find the point (x, y, z) on the surface that is closest to the origin (0, 0, 0). This can be done by finding the partial derivatives of f(x, y) and setting them equal to zero.
∂f/∂x = -y - 6 = 0
∂f/∂y = -x = 0
From the second equation, we can solve for x and get x = 0. Substituting this result into the first equation, we get -y - 6 = 0, which gives y = -6.
Substituting x = 0 and y = -6 into the equation of the surface, we can solve for z:
z^2 = 33 - (0)(-6) - 6(0)
z^2 = 33
Taking the square root of both sides, we get z = ±√33.
Therefore, there are two points on the surface that are equidistant from the origin: (0, -6, √33) and (0, -6, -√33).
To find the shortest distance, we can calculate the distance between each of these points and the origin using the distance formula:
d = sqrt(x^2 + y^2 + z^2)
For the first point (0, -6, √33), the distance is:
d = sqrt(0^2 + (-6)^2 + (√33)^2)
= sqrt(0 + 36 + 33)
= sqrt(69)
For the second point (0, -6, -√33), the distance is:
d = sqrt(0^2 + (-6)^2 + (-√33)^2)
= sqrt(0 + 36 + 33)
= sqrt(69)
Therefore, the shortest distance from the surface to the origin is sqrt(69).
What is the shortest distance from the surface xy+6x+z^2=33
to the origin?
1 answer