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

Answer:

The parametrization of the curve on the surface is

$c(t) = [x(t) , y(t), z(t)] \equiv [\frac{\sqrt{97} }{2} cost - 4 , \frac{\sqrt{97} }{2} sint + \frac{5}{2} , 5\frac{\sqrt{97} }{2} sint -8 \frac{\sqrt{97} }{2} cost +\frac{93}{2} ]$

Where

   $x = \frac{\sqrt{97} }{2} cost - 4$

    $y = \frac{\sqrt{97} }{2} sint + \frac{5}{2}$

$z = 5\frac{\sqrt{97} }{2} sint -8 \frac{\sqrt{97} }{2} cost +\frac{93}{2}$

Step-by-step 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

$(x + 4)^2 + (y - \frac{5}{2} )^2 = 2 +(\frac{5}{2} )^2 + 4^2$

$(x + 4)^2 + (y - \frac{5}{2} )^2 = \frac{97}{4}$

$(x + 4)^2 + (y - \frac{5}{2} )^2 = ( \frac{\sqrt{97} }{2} )^2$

representing the above equation in parametric form

$(x + 4) = \frac{\sqrt{97} }{2} cost$ , $(y -\frac{5}{2} ) = \frac{\sqrt{97} }{2} sin t$

$x = \frac{\sqrt{97} }{2} cost - 4$

$y = \frac{\sqrt{97} }{2} sint + \frac{5}{2}$

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

$z = 5[\frac{\sqrt{97} }{2} sint + \frac{5}{2}] -8[ \frac{\sqrt{97} }{2} cost - 4] +2$

$z = 5\frac{\sqrt{97} }{2} sint + \frac{25}{2} -8 \frac{\sqrt{97} }{2} cost + 32 +2$

$z = 5\frac{\sqrt{97} }{2} sint -8 \frac{\sqrt{97} }{2} cost +\frac{93}{2}$

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

$c(t) = [x(t) , y(t), z(t)] \equiv [\frac{\sqrt{97} }{2} cost - 4 , \frac{\sqrt{97} }{2} sint + \frac{5}{2} , 5\frac{\sqrt{97} }{2} sint -8 \frac{\sqrt{97} }{2} cost +\frac{93}{2} ]$