Question

Consider the paraboloid z=x2+y2. The plane 8x−5y+z−2=0 cuts the paraboloid, its intersection being a curve. Find "the natural" parametrization of this curve. Hint: The curve which is cut lies above a circle in the xy-plane which you should parametrize as a function of the variable t so that the circle is traversed counterclockwise exactly once as t goes from 0 to 2*pi, and the parameterization starts at the point on the circle with largest x coordinate. Using that as your starting point, give the parametrization of the curve on the surface.

c(t)=(x(t),y(t),z(t)), wherex(t)=y(t)=z(t)=

Answer 1

The parametrization of the curve on the surface is

`c(t) = [x(t) , y(t), z(t)] \equiv [(√(97) )/(2) cost - 4 , (√(97) )/(2) sint + (5)/(2) , 5(√(97) )/(2) sint -8 (√(97) )/(2) cost +(93)/(2) ]`

Where

`x =  (√(97) )/(2) cost - 4`

`y = (√(97) )/(2)  sint  + (5)/(2)`

`z = 5(√(97) )/(2) sint -8 (√(97) )/(2) cost +(93)/(2)`

Explanation:

From the question we are told that

The equation for the paraboloid is
`z = x^2 + y^2`

The equation of the plane is
`8x - 5y + z -2 = 0`

Form the equation of the plane we have that

`z = 5y -8x +2`

So

`x^2 + y^2 = 5y -8x +2`

=>
`x^2 + 8x + y^2 -5y = 2`

Using completing the square method to evaluate the quadratic equation we have

`(x + 4)^2 + (y - (5)/(2) )^2 = 2 +((5)/(2) )^2 + 4^2`

`(x + 4)^2 + (y - (5)/(2) )^2 = (97)/(4)`

`(x + 4)^2 + (y - (5)/(2) )^2 = ( (√(97) )/(2) )^2`

representing the above equation in parametric form

`(x + 4) = (√(97) )/(2) cost` ,
`(y -(5)/(2) ) = (√(97) )/(2) sin t`

`x = (√(97) )/(2) cost - 4`

`y = (√(97) )/(2) sint + (5)/(2)`

So from
`z = 5y -8x +2`

`z = 5[(√(97) )/(2) sint + (5)/(2)] -8[ (√(97) )/(2) cost - 4] +2`

`z = 5(√(97) )/(2) sint + (25)/(2) -8 (√(97) )/(2) cost + 32 +2`

`z = 5(√(97) )/(2) sint -8 (√(97) )/(2) cost +(93)/(2)`

Generally the parametrization of the curve on the surface is mathematically represented as

`c(t) = [x(t) , y(t), z(t)] \equiv [(√(97) )/(2) cost - 4 , (√(97) )/(2) sint + (5)/(2) , 5(√(97) )/(2) sint -8 (√(97) )/(2) cost +(93)/(2) ]`