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Problem 16 Easy Difficulty

(a) Eliminate the parameter to find a Cartesian equation of the curve.
(b) Sketch the curve and indicate with an arrow the direction in which the curve is traced as parameter increases.

$ x = \sqrt{t + 1} $, $ \quad y = \sqrt{t - 1} $

Answer

(a) $\frac{x^{2}}{2}-\frac{y^{2}}{2}=1, \quad x \geq \sqrt{2}, y \geq 0$
(b) See Graph

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Video Transcript

So this problem in particular, we're going to be looking at these curves defined by a parametric equations Eso What we were given is that X is equal to the square root of T plus one. And why is equal to the square root of T minus one? So with this in mind, we note that for t to be defined team must be greater than or equal to one. Um, which means that task is greater than or equal to for two. And why is greater than or equal to zero? So in this case, we want to solve for T in both equations, setting them equal to each other. So in this case, we get that X squared minus one equals T, which is going to equal Why squared plus one. So we ultimately end up getting that X squared minus y squared equals two or more similarly, X squared over two minus y squared over two equals one. Now, this right here, if you remember Con X, this is a hyper Bella centered at the origin. Um, so that is going to be our kind of equation right there or our elimination of the parameter. Next, we want to sketch the curve. Um, but since we know that, why is greater than or equal to zero If we graph this, it would look something like, uh, this right here. But since we're just focusing on why greater than or equal to zero and X greater than or equal to the square root of two, All we end up getting is this portion of the graph right here. So that would be our Parametric sketch.

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