## 17Calculus Parametric Equations - Slope & Tangent Lines

##### 17Calculus

On this page we discuss calculating slope and tangent lines of parametric equations.

To find the equation of a tangent line to a graph given by a set of parametric equations, we need to be able to find the slope by calculating the derivative $$dy/dx$$ using the parametric derivative $\frac{dy}{dx} = \frac{dy/dt}{dx/dt} \text{ where } dx/dt \neq 0$ For horizontal tangent lines, the slope $$dy/dx$$ is zero, so we need $$dy/dt = 0$$ and $$dx/dt \neq 0$$. For vertical tangent lines, the slope is undefined, which means that $$dx/dt = 0$$ when $$dy/dt \neq 0$$. In the case where both $$dx/dt = 0$$ and $$dy/dt = 0$$ at the same point, we need to handle that case separately, since nothing can be concluded from $$dy/dx = 0/0$$, which is indeterminate.

Once you have found the slope, you can easily find the equation of a tangent line. Go to the Tangent Lines page for more information.

Practice

Unless otherwise instructed, find the equation of the tangent line to these parametric curves at the given points.

$$x=4t$$, $$y=3t^2+2$$; $$t=2$$.

Problem Statement

Find the equation of the tangent line to the parametric curve $$x=4t$$, $$y=3t^2+2$$ at $$t=2$$.

Solution

### The Organic Chemistry Tutor - 3501 video solution

Log in to rate this practice problem and to see it's current rating.

$$x=t^2-5$$, $$y=t^2-4t$$; $$(-4,-3)$$

Problem Statement

Find the equation of the tangent line to the parametric curve $$x=t^2-5$$, $$y=t^2-4t$$ at $$(-4,-3)$$.

Solution

### The Organic Chemistry Tutor - 3502 video solution

Log in to rate this practice problem and to see it's current rating.

$$x=8\cos\theta$$, $$y=6\sin\theta$$; $$\theta = \pi/3$$

Problem Statement

Find the equation of the tangent line to the parametric curve $$x=8\cos\theta$$, $$y=6\sin\theta$$ at $$\theta = \pi/3$$.

Solution

### The Organic Chemistry Tutor - 3503 video solution

Log in to rate this practice problem and to see it's current rating.

$$x=2t^2+1$$, $$y=3t^3+2$$; $$t=1$$.

Problem Statement

Find the equation of the tangent line to the parametric curve $$x=2t^2+1$$, $$y=3t^3+2$$ at $$t=1$$.

Solution

### Krista King Math - 54 video solution

video by Krista King Math

Log in to rate this practice problem and to see it's current rating.

$$x=3(t-\sin(t))$$, $$y=3(1-\cos(t))$$; $$t=\pi/2$$.

Problem Statement

Find the equation of the tangent line to the parametric curve $$x=3(t-\sin(t))$$, $$y=3(1-\cos(t))$$ at $$t=\pi/2$$.

Solution

### Krista King Math - 55 video solution

video by Krista King Math

Log in to rate this practice problem and to see it's current rating.

$$x=t\cos(t)$$, $$y=t\sin(t)$$; $$t=\pi$$

Problem Statement

Find the equation of the tangent line to the parametric curve $$x=t\cos(t)$$, $$y=t\sin(t)$$ at $$t=\pi$$.

Solution

### Krista King Math - 467 video solution

video by Krista King Math

Log in to rate this practice problem and to see it's current rating.

$$x=3t^2-t$$, $$y=\sqrt{t}$$; $$t=4$$.

Problem Statement

Find the equation of the tangent line to the parametric curve $$x=3t^2-t$$, $$y=\sqrt{t}$$ at $$t=4$$.

Solution

### PatrickJMT - 1371 video solution

video by PatrickJMT

Log in to rate this practice problem and to see it's current rating.