Question

Find the shortest distance from origin to the surface xyz2 = 2.​

Answer 1

z2=2/(xy) d2=x2+y2+z2=x2+y2+2/(xy) Partial derivative relative to x : 2x-2/(x2y)=0 Partial derivative relative to y : 2y-2/(xy2)=0 Solving the equations y*(x3)=x*(y3)=1 leads to : x=y=1 then z=√2 min.distance d=2Read more on Sarthaks.com - https://www.sarthaks.com/523480/shortest-distance-from-origin-to-xyz-2-2

Answer 2

Answer:

The shortest distance from origin to the given curve is 2 units

Step-by-step explanation:

The given geometrical surface in the Cartesian co-ordinate system is

$xyz^{2}=2$

We can rewrite the curve in the following way

$xyz^{2}=2= > z^{2}=\frac{2}{xy}$

Consider any arbitrary point on this surface as P(x,y,z).This point can also be represented as $P(x,y,\sqrt{\frac{2}{xy} } )$

Now distance from origin to the above mentioned point on given surface is

$d=\sqrt{x^{2} +y^{2} +\frac{2}{xy} }$ squaring on both the sides we get,

$d^{2}={x^{2} +y^{2} +\frac{2}{xy} }$ as the distance is minimum,the partial derivative of this distance must be zero.Hence,

Partial derivative with respect to variable x is $\frac{2x-2}{x^{2} y} =0== > 2x-2=0= > x=1$

Similarly,Partial derivative with respect to variable y is$\frac{2y-2}{x^{2} y} =0== > 2y-2=0= > y=1$

As x and y are found the z-coordinate is $\sqrt{\frac{2}{1\times1} } =\sqrt{2}$

Hence,the point at shortest distance is $P(1,1,\sqrt{2} )$

Now,the distance of this point from the origin is $d=\sqrt{1^{2} +1^{2} +2} =2 units$

Therefore,The shortest distance from origin to the given curve is 2 units.

#SPJ3