Find eqn for "normal" line to the tangent- folium

In summary, the graph of a folium with equation ##2x^3+2y^3-9xy=0## has a tangent line at (2,1) and a normal line to the tangent line at (2,1).
  • #1

opus

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Homework Statement


Back for more questions. This section has been pretty tricky.

The graph of a folium with equation ##2x^3+2y^3-9xy=0## is given.

Find the equation for the tangent line at the pont ##(2,1)##

Find the equation of the normal line to the tangent line in the last question at the point (2,1).

Homework Equations

The Attempt at a Solution



From part a, the tangent line at the point ##(2,1)## is ##y=\frac{5}{4}x-\frac{3}{2}##

Now by "normal" line, I assume that the question is talking about the folium itself.
But since we're talking about the line at that point, wouldn't the equation for the normal line at that exact point just be the tangent line? Otherwise, how could we determine how far out to choose our x values?
 
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  • #2
A normal is something perpendicular. So you are looking for the line through ##(2,1)## which is perpendicular to the tangent. Do you know what the slope of such a perpendicular line is, given that the tangent has slope ##\frac{5}{4}\,?##
 
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  • #3
From geometry that would just be the reciprocal right?
 
  • #4
opus said:
From geometry that would just be the reciprocal right?
In a way, yes. But it is negative, since ##\frac{5}{4}>0##. It is as if ##x## and ##y## axis were swapped. The formula for an angle between two vectors ##\vec{v}=(v_1,v_2)\; , \;\vec{w}=(w_1,w_2)## is
$$
\cos \sphericalangle (\vec{v},\vec{w}) = \dfrac{\vec{v}\cdot \vec{w}}{|\vec{v}|\cdot |\vec{w}|}
$$
where the product is ##\vec{v}\cdot \vec{w} = v_1w_1+v_2w_2## and ##|\vec{v}|=\sqrt{v_1^2+v_2^2}##.

Here we have ##\vec{v}=(1,\frac{5}{4})## and the cosine of ##90°## is zero.
 
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  • #5
So then we would have the equation for the normal line is ##y=\frac{-4}{5}x+\frac{13}{5}##?
 
  • #6
opus said:
So then we would have the equation for the normal line is ##y=\frac{-4}{5}x+\frac{13}{5}##?
Yes - as long as part a) is correct, which I haven't checked.
 
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  • #7
Ok cool stuff! Thanks again!
 

What is a "normal" line to the tangent-folium?

A "normal" line to the tangent-folium is a line that is perpendicular to the tangent line at a specific point on the curve. It intersects the tangent line at the point of tangency and has a slope that is the negative reciprocal of the slope of the tangent line.

How do you find the equation for the normal line to the tangent-folium?

The equation for the normal line to the tangent-folium can be found using the following steps:1. Find the slope of the tangent line at the point of tangency.2. Calculate the negative reciprocal of the tangent line's slope.3. Use the point-slope form of a line to write the equation, using the point of tangency and the negative reciprocal slope.4. Simplify the equation to get the final form of the normal line's equation.

What is the significance of the normal line in relation to the tangent-folium?

The normal line is significant because it provides important information about the behavior of a curve at a specific point. It is perpendicular to the tangent line, which means it is orthogonal to the curve at that point. This can be useful in understanding the geometry and properties of the curve.

Can the equation for the normal line to the tangent-folium be used to find other points on the curve?

Yes, the equation for the normal line can be used to find other points on the curve. This is because the normal line and the curve intersect at the point of tangency. By finding the x and y coordinates of the point of tangency, we can substitute them into the equation for the normal line and solve for the corresponding y or x value on the curve.

Are there any real-world applications of finding the equation for the normal line to the tangent-folium?

Yes, there are many real-world applications of finding the equation for the normal line to the tangent-folium. For example, in engineering and physics, the normal line is used to determine the forces acting on an object at a specific point. In computer graphics, the normal line is used to create realistic 3D images. It also has applications in navigation, where the normal line can be used to determine the direction of travel on a curved path.

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