What is the shortest distance from the surface xy+6x+z^2=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.
First, let's rewrite the equation of the surface in the standard form of a plane:
xy + 6x + z^2 = 33
Rearranging the terms, we get:
xy + 6x - 33 + z^2 = 0
Now, we can find the direction vector of the plane by taking the gradient of the surface equation:
∇(xy + 6x - 33 + z^2) = <∂/∂x, ∂/∂y, ∂/∂z>(xy + 6x - 33 + z^2)
= <y + 6, x, 2z>
Next, we need to find a point on the surface. Since we want to find the distance to the origin, we can choose a point on the surface with coordinates (0, 0, ±√33). Let's choose (0, 0, √33).
Now, we can calculate the shortest distance from the origin to the surface using the formula:
d = |Ax + By + Cz + D| / √(A^2 + B^2 + C^2)
where A, B, and C are the direction vector components, and (x, y, z) is any point on the surface.
Plug in the values:
A = y + 6 = 0 + 6 = 6
B = x = 0
C = 2z = 2√33
D = -33
(x, y, z) = (0, 0, √33)
d = |6(0) + 0 + 2√33(√33) - 33| / √(6^2 + 0^2 + (2√33)^2)
= |66 - 33| / √(36 + 132)
= 33 / √168
Simplifying the radical, we get:
d = 33 / (2√42)
So, the shortest distance from the surface xy + 6x + z^2 = 33 to the origin is 33 / (2√42) units.
To find the shortest distance from the surface to the origin, we need to find the perpendicular distance between them.
The surface equation is given by: xy + 6x + z^2 = 33.
To find the distance between a point and a surface, we need to take the dot product between the normal vector of the surface and a vector connecting the point on the surface to the origin.
First, let's find the normal vector of the surface. The equation can be rewritten in the standard form as follows:
z^2 + xy + 6x = 33.
Comparing this with the general equation for a surface in 3D, Ax + By + Cz + D = 0, we have A = 1, B = x, C = 0, and D = xy + 6x - 33.
Therefore, the normal vector is N = (A, B, C) = (1, x, 0).
Next, we need a vector connecting a point on the surface to the origin. We'll choose a point on the surface with coordinates (x, y, z). The vector from this point to the origin is given by V = (-x, -y, -z).
Now, we can find the dot product between the normal vector and the vector connecting the point to the origin:
N · V = (1, x, 0) · (-x, -y, -z) = -x - xy.
To find the shortest distance, we need to minimize the magnitude of the dot product N · V. This is equivalent to finding the minimum value for -x - xy.
To find this minimum value, let's differentiate -x - xy with respect to x and y, and set the derivatives equal to zero:
d/dx (-x - xy) = -1 - y = 0 (Equation 1),
d/dy (-x - xy) = -x = 0 (Equation 2).
From Equation 2, we get x = 0. Substituting this into Equation 1, we have -1 - y = 0, which gives y = -1.
Therefore, the coordinates of the point on the surface closest to the origin are (0, -1, z).
To find the value of z, we substitute these values into the surface equation:
0(-1) + 6(0) + z^2 = 33,
z^2 = 33,
z = ±√33.
Since z cannot be negative (as it represents a distance), the value of z is z = √33.
Finally, to find the shortest distance, we calculate the magnitude of the vector connecting the point on the surface to the origin:
Distance = √[(-0)^2 + (-1)^2 + (√33)^2],
Distance = √(1 + 1 + 33),
Distance = √35.
Therefore, the shortest distance from the surface xy + 6x + z^2 = 33 to the origin is √35 units.