Question

points on the surface Z^2-xy=1 nearest to origin​

Answer 1

Answer:

The point (x, x,−1) must satisfy g = 0. Thus, −1 − x2 − 1=0 That is, x2 = −2. ... That is, the point on the surface z = xy + 1 closest to the origin is (0,0,1).

Answer 2

Concept

A surface could be a set of dimension two; this suggests that a moving point on a surface may move in two directions.

Given

The given surface is $z^2-xy=1$.

Find

We have to seek out the points nearest to the origin

Solution

Let the points be $(x,y,z)$ and thatwe have $z^2-xy=1$.

So, we will rewrite it as $z=\sqrt{xy\pm 1}$.

Thus, the gap $D$ from the origin is

$\begin{aligned}D^2&=x^2+y^2+z^2\\ D^2&=x^2+y^2+xy+1\end$

If $D$ is minimum $D^2$ is additionally minimum.

So,

$\begin{aligned}\frac{\partial D^2}{\partial x}&=2x+y=0\\ \frac{\partial D^2}{\partial y}&=2y+x=0\end$

Therefore, $x=0,y=0,z=\pm 1$ be the points that are nearest to origin are $(0,0,1),(0,0,-1).$

Hence, the solution is $(0,0,1),(0,0,-1).$

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