What is the shortest distance from the surface xy+3x+z2=9xy+3x+z2=9 to the origin?

Shortest distance is (2,1,1)

Step-by-step explanation:

Using the formula for distance. D² = x² + y² + z²

Minimizing D² is just as valid as minimizing D.

Now, let's rearrange the original equation to get z² = 9 - xy - 3x

Let's put this into the equation for D² to obtain;

D² = x² + y² + 9 - xy - 3x

Taking the partial derivatives with respect to x and y and set them equal to 0 to get the minimum, we have;

dD²/dx = 2x - y - 3 = 0

Also;

dD²/dy = 2y - x = 0

Solve dD²/dy for x, we have:

x = 2y

Put this into the first equation to obtain: 2(2y) - y = 3

This gives ; 3y = 3

So y = 1 and x = 2

Let's plug in y = 1 and x = 2 into

z² = 9 - xy - 3x

Thus gives;

z² = 9 - (1 x 2) - (3 x 2)

z² = 9 - 2 - 6

z² = 1

And z = 1

So the closest point is (2, 1, 1)

8(-10x+3.5y-7)

8×-10x + 8×3.5y + 8×-7

-80x + 28y - 56

therefore

-80x+28y-56 is the correct answer