Parametric Curves: Tangent Lines

In summary, the conversation discusses finding equations of tangents to a curve that pass through a given point. The equation y=x-1 is found for the part of the curve above the x-axis, but the equation for the part below is still unknown. The suggestion is made to use the general equation for the tangent line and plug in the given point to find a couple of cubic equations. It is noted that t=1 is a solution, but the second equation is still needed. Eventually, the second equation is found with the help of others.
  • #1

Homework Statement



Find equations of the tangents to the curve x=3t^2+1, y=2t^3+1 that pass through the point (4,3).

The Attempt at a Solution



I was able to find the equation y=x-1 as a tangent line through the point (4,3) for the part of the curve above the x-axis since (4,3) is on the curve.

However, I do not know how to find the equation of the tangent for the part of the curve below the x-axis since I do not know what point on the curve the tangent passes through and therefore do not know the t value.

Any help is appreciated, thanks
 
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  • #2
Write down the equation for the tangent line to the curve at general value of t. Now put in the condition (4,3) is on the line. You should get a couple of cubic equations. You already know t=1 is a solution. Is there another?
 
  • #3
Well, you get a cubic equation. Knowing that t= 1 is a solution makes it easy to solve. Dick's suggestion is right on the money.
 
  • #4
Duh, right. Make that 'one' cubic.
 
  • #5
thanks a lot for the help, i was able to find the second equation, thanks
 
  • #6
So what *is* that second equation? I'm stuck!
 

1. What are parametric curves?

Parametric curves are mathematical representations of curves or lines in a coordinate system that are defined by a set of equations, known as parametric equations. These equations use one or more parameters to describe the coordinates of points on the curve or line.

2. How are tangent lines related to parametric curves?

Tangent lines are lines that touch a curve or line at a single point, and have the same slope as the curve at that point. For parametric curves, the tangent line is defined by the derivative of the parametric equations with respect to the parameter.

3. How do you find the tangent line to a parametric curve?

To find the tangent line to a parametric curve at a specific point, you first need to find the derivative of the parametric equations with respect to the parameter. Then, substitute the coordinates of the point into the derivative to find the slope of the tangent line. Finally, use the point-slope formula to write the equation of the tangent line.

4. What is the significance of tangent lines in parametric curves?

Tangent lines play an important role in understanding the behavior of parametric curves. They provide information about the slope and direction of the curve at a given point, which can be used to determine the concavity and curvature of the curve. Tangent lines also have applications in optimization problems and motion analysis.

5. Can parametric curves have more than one tangent line at a point?

Yes, parametric curves can have more than one tangent line at a point. This occurs when the curve has a sharp turn or corner, known as a cusp. In these cases, the curve has two distinct tangent lines at the point of the cusp, each with a different slope.

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