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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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) .`

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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