Find the equations of the tangent lines at the point where the curve crosses itself. x=t^2-t,

Find the equations of the tangent lines at the point where...

Key Concepts

Parametric Curves

Parametric curves represent the coordinates of points on a curve as functions of a separate parameter. This representation allows for analysis of curves that might loop or intersect themselves and is particularly useful when the curve does not represent y explicitly as a function of x.

Self-Intersection Points

A self-intersection is a point on a curve where two or more distinct parameter values yield the same coordinates. Recognizing self-intersections is important because such points often have multiple tangent directions, each corresponding to a different branch of the curve.

Differentiation of Parametric Equations

When dealing with curves defined parametrically, the derivative dy/dx is obtained by dividing the derivative of y with respect to the parameter by the derivative of x with respect to the parameter. This process is essential for finding the slope of the tangent line at any particular point along the curve.

Tangent Lines

A tangent line to a curve at a particular point is the line that best approximates the curve near that point. In the context of parametric curves, once the slope is determined from the derivatives, the tangent line can be formulated using the point-slope form. At a self-intersection, multiple tangent lines may exist, each corresponding to the different parameter values that arrive at the same point.

Transcript

00:01 In this question, we want to find the equation of tangent at the point where the graph crosses itself.

00:07 So first of all, we need to graph this pair of parametric function to see where the graph actually crosses itself.

00:18 Two, if we're talking about tangent, we need to find the dy, dx.

00:22 So the d y, dx in terms of parametric, will be d .y over dt divided by dx over dt.

00:31 And third, to find the tangent, you have to use the equation formula, y minus y1, equals to m, x minus x1, where x1, y1 is the point where the graph crosses itself, and m is the gradient at that point.

00:49 Now, let's graph the graph.

00:52 So if you were to use any graphing utility to graph the parametric graph, you'll realize that the graph looks a little bit like, what i'm going to draw here, it may not be very nice.

01:05 So it looks something like this.

01:08 Now, it will cross itself at this point to 1.

01:13 Okay? so i will want to find the t value as it crosses at the point 21.

01:21 So let's see.

01:23 It's easier to set 2 to the x over here to find the t.

01:30 So here we can find it.

01:33 I'm going to set when x.

01:35 Equals to 2, 2 is equal to t squared minus t.

01:43 So you'll get a quadratic, right? quadratic will be t squared minus t minus 2 equals 0...