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1a) Sketch and identify the surface `4y^2-x^2-(z-2)^2=4` ` ` b)Find a vector function...

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haqc | eNoter

Posted October 22, 2013 at 6:14 AM via web

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1a) Sketch and identify the surface

`4y^2-x^2-(z-2)^2=4`

` `

b)Find a vector function that represents the curve of intersection of the surfaces `x^2+y^2=8and z=3xy+7`

``

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aruv | High School Teacher | Valedictorian

Posted October 22, 2013 at 8:23 AM (Answer #1)

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Let us assume

`x=m`  then

`m^2+y^2=8`

`y^2=8-m^2` 

`y=+-sqrt(8-m^2)`                 (i)

Substitute x and y in the equation of the curve  z=3xy+7 i.e.

`z=3msqrt(8-m^2)+7 and z=-3msqrt(8-m^2)+7`

Thus vector functions are

`(m,sqrt(8-m^2),3msqrt(8-m^2)) and (m,-sqrt(8-m^2),-3msqrt(8-m^2)+7) .`

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tiburtius | High School Teacher | (Level 3) Associate Educator

Posted October 22, 2013 at 12:55 PM (Answer #2)

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a)

As you can see from the picture below this hyperboloid of two sheets. This is easily seen even without picture because general formula for hyperboloid with two sheets is

`x^2/a^2-y^2/b^2-z^2/c^2=1`

where `a,b,c in RR`  ``

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