Intersection of Two Curves: Do They Meet?

In summary, Mark44 tried to solve an equation that represented the distance between two geometric shapes but was unsuccessful. He suggested trying a different approach, such as finding the point on the parabola closest to the origin.
  • #1
PensNAS
12
0

Homework Statement



Show that these curves do not intersect.

z=(1/a)(a-y)^2
y^2+z^2=a^2/4

Where a is the radius of the circle and other shape.

Homework Equations



There aren't any.

The Attempt at a Solution



I tried setting them equal to each other but got the equation, which doesn't simplify into anything solve able.

a^2/4-y^2=a^2-4ay+6y^2-4y^3/a+y^4/a^2
 
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  • #2
PensNAS said:

Homework Statement



Show that these curves do not intersect.

z=(1/a)(a-y)^2
y^2+z^2=a^2/4

Where a is the radius of the circle and other shape.

Homework Equations



There aren't any.

The Attempt at a Solution



I tried setting them equal to each other but got the equation, which doesn't simplify into anything solve able.

a^2/4-y^2=a^2-4ay+6y^2-4y^3/a+y^4/a^2

I think that the easiest way to do this is to start with a sketch of the two curves. The 2nd equation is of a circle - what is its radius?

What sort of figure does the first equation represent?
 
  • #3
The first equation looks similar to z=1/x. The radius of the circle is a/2. I've already sketched out the equations.
 
  • #4
PensNAS said:
The first equation looks similar to z=1/x.
No, not at all. The 1/a part is just a constant. The variable part, (a - y)2, is what's important.
PensNAS said:
The radius of the circle is a/2. I've already sketched out the equations.
That's correct for the circle's radius.
 
  • #5
It is a decreasing, concave curve is what I should have said. How does one show that there is not a solution for both equations?
 
  • #6
PensNAS said:
It is a decreasing, concave curve is what I should have said.
It is concave up, yes, but it's decreasing on part of its domain and increasing on the other part.

What geometric figure is y = K(a - x)2? (I changed the variables so that you might recognize what this is more easily.)
PensNAS said:
How does one show that there is not a solution for both equations?
Like I said before, a sketch of both curves is a good start. So far, you have only one of them.
 
  • #7
It is a parabola. The book gives us the graph of both functions, but limits it to the first quadrant. I should have soda that too, my bad.
 
  • #8
You could try this: Write a function that represents the distance between an arbitrary point on the circle and one on the left side of the parabola. Find the minimum value of that function. If the minimum value is greater than zero, the two curves don't intersect.

Trying to solve the equation that you found seems like a dead end to me.
 
  • #9
Am I wrong in thinking the formula for the distance between the functions is the top curve minus the bottom curve?
 
  • #10
PensNAS said:
Am I wrong in thinking the formula for the distance between the functions is the top curve minus the bottom curve?

Yes, you're wrong. That would give you the vertical distance between the two curves, which isn't the same as the distance I described.

A slightly different approach would be to find the point on the parabola that's closest to the origin. Then the line segment from the origin to the parabola would intersect the circle at some point, and you could easily find the distance between the circle and the parabola.

I'm sort of making this up as I go along, but I think this is a workable way to go.
 
  • #11
Offtopic, but do you happen to be in Pensacola?
 
  • #12
Mark44 said:
Yes, you're wrong. That would give you the vertical distance between the two curves, which isn't the same as the distance I described.

A slightly different approach would be to find the point on the parabola that's closest to the origin. Then the line segment from the origin to the parabola would intersect the circle at some point, and you could easily find the distance between the circle and the parabola.

I'm sort of making this up as I go along, but I think this is a workable way to go.

Since the second equation is a circle centered at the origin, the distance from a point on the parabola to the origin is going to give you a fourth order equation, not too much different from what you get trying to intersect them. It might pay to be more clever. If pensNAS has got a good picture, think about drawing a tangent line to the circle that has the circle on one side and the parabola on the other. Then if you write down an equation for the tangent line and show it doesn't intersect the parabola, you would be done.
 
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  • #13
Thanks for the help! I actually went and talked to my professor, since this is part of a statics problem and not calculus. He was okay with just plotting a few points around where the curves get closest.

@Mark44, I'm from Pensacola.
 
  • #14
PensNAS said:
Thanks for the help! I actually went and talked to my professor, since this is part of a statics problem and not calculus. He was okay with just plotting a few points around where the curves get closest.

@Mark44, I'm from Pensacola.

Ok. For the record, I think it's pretty easy to show that the tangent line to the circle passing through x=a/(2sqrt(2)), y=a/(2sqrt(2)) doesn't intersect the parabola. Since the two curves are on different sides of that line, they can't intersect.
 

FAQ: Intersection of Two Curves: Do They Meet?

What is the intersection of two curves?

The intersection of two curves refers to the point or points where two different curves on a graph or plot meet or cross each other. This point represents the values of the independent variable(s) where the two curves have the same value.

How do you find the intersection of two curves?

To find the intersection of two curves, you need to solve the equations of the two curves simultaneously. This can be done by graphically plotting the two curves and visually identifying the point(s) of intersection, or algebraically by setting the equations equal to each other and solving for the values of the independent variable(s) that satisfy both equations.

Can two curves intersect at more than one point?

Yes, two curves can intersect at more than one point. This occurs when the two curves have multiple points where they have the same value of the independent variable(s).

What does it mean if two curves do not intersect?

If two curves do not intersect, it means that they do not have any points where they have the same value of the independent variable(s). This could indicate that the two curves are parallel or do not have any common points of interest.

How can the intersection of two curves be used in real-life applications?

The intersection of two curves can be used in various real-life applications such as in economics to determine the equilibrium point of supply and demand curves, in physics to find the point of collision between two moving objects, and in engineering to analyze the behavior of complex systems with multiple variables. It is a useful tool for understanding the relationship between two variables and for making predictions based on their intersection point.

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