Angle Between a Parabola and a Vertical Line

Question

I am trying to find the angle between a parabola $y=-0.000314x^2+0.3716x$ and a vertical line $x=738$.

I found that I have to use this formula: $$\tan \theta=\frac{m_2-m_1}{1+m_1.m_2}$$but I'm not sure how I could use this as $X=738$ has an undefined slope. Any help on how to get started would be great! :)

Answer 1

The angle $\theta$ between a line of slope $m$ and the $x$-axis satisfies $\tan \theta = m$. So in your case the angle between $x_0=738$ and the $x$-axis is just $90°$ or $\pi/2$. Now you just have to find the slope of the tangent of your parabola, then find it's angle to the $x$-axis, and calculate the difference to the $90°$.

The slope of th tangent of the parabola at $x_0=738$ is just the value of the derivative $f'(x_0)$ of the function $f(x) = -0.000314x^2+0.3716x$ at $x_0$.

Answer 2

The point of intersection between the line $x=738$ and the parabola $f(x)=-0.000314x^2+0.3716x$ is the point $M(738,103.222584)$.

At this point M, the slope of the line tangent to $f(x)$ is $f'(x)$ at $x=738$, which is $-0.091864$.

$\tan{\theta}=\frac{m_{2}-m_{1}}{1+m_{1}m_{2}}=\frac{1-\frac{m_{1}}{m_{2}}}{\frac{1}{m_{2}}+m_{1}}$

As $m_{2}\rightarrow{\infty}$, $\frac{1}{m_{2}}\rightarrow{0}$. So, the above expression reduces to $\frac{-1}{m_{1}}=\frac{1}{-0.091864}=-10.885657$

Answer 3

Did someone tell you that you "have to" use the formula $\tan \theta=\frac{m_2-m_1}{1+m_1 \cdot m_2}$? That would be a bit odd, since (as you observed) the formula is not really suited to this problem.

Remember that if you have a line of slope $m_1$ that makes an angle $\theta_1$ with the $x$-axis, the relationship between that line's slope and angle is $$ \tan\theta_1 = m_1. \tag1 $$ Now consider another line of slope $m_2$ and angle $\theta_2$ with the axis: $$ \tan\theta_2 = m_2. \tag2 $$ The angle between those lines, measured from the "positive" direction of the first line counterclockwise to the "positive" direction of the second line, is the difference between the two angles $\theta_2$ and $\theta_1$: $$ \theta = \theta_2 - \theta_1. \tag3 $$ That is the basic set of formulas for determining the angle between two lines. It also easily adapts to the case where one or both of the lines is vertical (and therefore does not have a slope): the angle such a line makes with the $x$-axis is $\frac\pi2$, that is, $90$ degrees. For example, if the first line were vertical, we wouldn't use Equation $(1)$ at all, but would simply set $\theta_1 = \frac\pi2$.

It happens that for two lines that both have defined slopes, there is a useful application of the angle difference formula for tangents: $$ \tan(\theta_2 - \theta_1) = \frac{\tan\theta_2 - \tan\theta_1}{1 + \tan\theta_1 \cdot \tan\theta_2}. $$ The left-hand side of this is just $\tan\theta$ (according to Equation $(3)$), while the right-hand side is $\frac{m_2-m_1}{1+m_1 \cdot m_2}$ (using Equations $(1)$ and $(2)$), and that's how we get $$\tan \theta=\frac{m_2-m_1}{1+m_1 \cdot m_2}. \tag4$$

In the cases where this formula applies, (when the slopes of both lines are defined), this formula allows you to find the angle $\theta$ by using the inverse tangent function just once, rather than twice (as you would have to if you used Equations $(1)$ and $(2)$ instead). In a computer program, the inverse tangent is generally a much more expensive thing to do than just multiplying or dividing numbers, so Equation $(4)$ can help a program run faster. But it is still only a (sometimes) useful formula, not the defining formula for the angle between two lines or curves.