At what points does the curve r(t) = t \space i + (9t - t^2)\space k intersect the paraboloid z =...

Question:

At what points does the curve {eq}r(t) = t \space i + (9t - t^2)\space k {/eq} intersect the paraboloid {eq}z = x^2 + y^2 {/eq}?

Parametric Curve:

A parametric curve in the zx plane is described by the set of parametric equations

{eq}x=x(t) \\ z=z(t) {/eq}

where t is the parameter.

The curve can be expressed in terms of a Cartesian equation by eliminating the parameter t,

for instance evaluating the parameter t from one equation and plugging its expression in the other equation.

Answer and Explanation:

We are given the parametric curve in the zx plane

{eq}\mathbf r(t) = t \mathbf i+ (9t - t^2) \mathbf k. {/eq}

The curve is expressed in cartesian form as follows

{eq}t =x \\ y=0 \\ \Rightarrow z = 9x -x^2. {/eq}

The intersection points between the curve and the paraboloid

{eq}z = x^2 + y^2 {/eq}

are found matching the expressions of the z-coordinates

{eq}z = z \Rightarrow 9x -x^2 = x^2 \\ \displaystyle \Rightarrow 2x^2 - 9x = 0 \\ \displaystyle \Rightarrow 2x^2 - 9x = 0 \\ \displaystyle \Rightarrow x=0, \; y=0\; z=0 \;\; and \\ \displaystyle x= \frac{9}{2},\:y=0\; z = \frac{81}{4}. {/eq}


Learn more about this topic:

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Graphs of Parametric Equations

from Precalculus: High School

Chapter 24 / Lesson 5
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