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(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} $

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

02:02

Wen Zheng

Calculus 2 / BC

Chapter 10

Parametric Equations and Polar Coordinates

Section 1

Curves Defined by Parametric Equations

Parametric Equations

Polar Coordinates

Campbell University

Baylor University

Boston College

Lectures

16:57

In mathematics, a graph is a representation of a set of objects where some pairs of the objects are in some sense "related". The objects correspond to mathematical abstractions called "vertices" or "nodes", and the relations between them are represented by mathematical abstractions called "edges" or "arcs". The basic notion of a graph was developed by the 17th-century French mathematician Pierre de Fermat, and the term "graph" was coined by the 19th-century mathematician James Joseph Sylvester. The more general mathematical concept of a graph "in which any kind of relation between elements of the set is expressed as an edge, is called a network" (Kolmogorov, "1956, p. 111"). In other words, an undirected graph is a graph in which the edges have no direction associated with them. The most familiar examples of graphs are the graphs of equations. In general, the vertices of a graph can represent concepts and the edges can represent real-valued functions on the concepts, so one can speak of the graph as a function's graph or of the edge as a function's edge.

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Polar coordinates are a two-dimensional coordinate system that specifies a point in terms of distance from a reference direction (the pole) and angle from a reference direction (the polar axis).

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(a) Eliminate the paramete…

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(a) Sketch the curve by us…

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Sketching a Curve by Elimi…

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a) Eliminate the parameter…

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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