# Find the points on the surface y^2 = 9 + xz that are closest to the origin.

This question aims to learn the basic methodology for optimizing a mathematical function (maximizing or minimizing).

Critical points are the points where the value of a function is either maximum or minimum. To calculate the critical point(s), we equate the first derivative’s value to 0 and solve for the independent variable. We can use the second derivative test to find maxima/minima. For the given question, we can minimize the distance function of the desired point from the origin as explained in the below answer.

Given:

$y^{ 2 } \ = \ 9 \ + \ x \ z$

Let $( x, \ y, \ z )$ be the point that is nearest to the origin. The distance of this point from the origin is calculated by:

$d = \sqrt{ x^{ 2 } + y^{ 2 } + z^{ 2 } }$

$\Rightarrow d^{ 2 } = x^{ 2 } + y^{ 2 } + z^{ 2 }$

$\Rightarrow d^{ 2 } = x^{ 2 } + 9 + x z + z^{ 2 }$

To find this point, we simply need to minimize this $f(x, \ y, \ z) \ = \ d^{ 2 }$ function. Calculating the first derivatives:

$f_x = 2x + z$

$f_z = x + 2z$

Finding critical points by putting $f_x$ and $f_z$ equal to zero:

$2x + z = 0$

$x + 2z = 0$

Solving the above system yields:

$x = 0$

$z = 0$

Consequently:

$y^{ 2 } = 9 + xz = 9 + (0)(0) = 0$

$\Rightarrow = y = \pm 3$

Hence, the two possible critical points are $(0, 3, 0)$ and $(0, -3, 0)$. Finding the second derivatives:

$f_{xx} = 2$

$f_{zz} = 2$

$f_{xz} = 1$

$f_{zx} = 1$

Since all second derivatives are positive, the calculated critical points are at  a minimum.

## Numerical Result

Points Closest to the origin = $(0, 0, 5)$ and $(0, 0, -5)$

## Example

Find the points on the surface  $z^2 = 25 + xy$ nearest to the origin.

Here, the distance function becomes:

$d = \sqrt{ x^{ 2 } + y^{ 2 } + z^{ 2 } }$

$\Rightarrow d^{ 2 } = x^{ 2 } + y^{ 2 } + z^{ 2 }$

$\Rightarrow d^{ 2 } = x^{ 2 } + y^{ 2 } + 25 + xy$

Calculating first derivatives and equating to zero:

$f_x = 2x + y \Rightarrow 2x + y = 0$

$f_y = x + 2y \Rightarrow x + 2y = 0$

Solving the above system yields:

$x = 0 \text{and} y = 0$

Consequently:

$z^{ 2 } = 25 + xy = 25$

$\Rightarrow = z = \pm 5$

Hence, the two possible critical points are $(0, 3, 0)$ and $(0, -3, 0)$. Finding the second derivatives:

$f_{xx} = 2$

$f_{yy} = 2$

$f_{xy} = 1$

$f_{yx} = 1$

Since all second derivatives are positive, the calculated critical points are at a minimum.

Points Closest to the origin = $(0, 0, 5)$ and $(0, 0, -5)$