Circle Inscribed in a Parabola?

In summary: The equation of a parabola is y=x^2. The region above the parabola has the same curvature as y=x^2 at the origin. The radius of curvature is given by r=v^2/g.
  • #1
DakMasterFlash
1
0

Homework Statement



Find the largest circle centered on the positive y-axis which touches the origin and which is above y=x^2

Homework Equations



equation of a circle: r^2=(x-a)^2+(y-b)^2
equation of a circle centered on the y-axis: x^2+(y-b)^2=r^2
equation of a parabola: y=x^2

The Attempt at a Solution



For the life of me, I couldn't figure out how to do this problem. I "eyeballed" the graphs and realized that a circle with a radius of 0.5 perfectly fits the parameters required by the problem statement, but I have no idea how to do this mathematically.

A radius of 0.5 yields the equation:
x^2+(y-0.5)^2=0.5^2
x^2+y^2-y+0.25=0.25
x^2+y^2-y=0

I've tried (rather aimlessly) taking the derivative of this function, yielding:
2x+2yy'-1y'=0

However, I realize that I should use the generic equation for a circle, which gives me:
x^2+(y-b)^2=r^2
x^2+y^2-yb+b^2=r^2

As I'm sure you all can tell, I'm not making much progress as to an actual solution to this problem.
I was hoping that you guys could give me some direction and a hint or two.

Any help would be appreciated! Thanks!
 
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  • #2
DakMasterFlash said:
equation of a circle: r^2=(x-a)^2+(y-b)^2
equation of a circle centered on the y-axis: x^2+(y-b)^2=r^2
So far so good. You are given one additional constraint: the circle must touch the origin. What does that tell you about how ##r## and ##b## are related to each other?

equation of a parabola: y=x^2
OK, can you write down an inequality that describes the region above the parabola?
 
  • #3
Another way to think about it is that you are looking for the 'osculating circle' to y=x^2 at (0,0). It will have the same curvature as y=x^2 at the origin. y'' must be the same for the two curves there.
 
  • #4
I have some other kind of approach which uses the concept of projectiles in physics (I know it sounds weird but it works :approve:). Here goes the solution:

Let us assume the parabola to be y= -x^2. I simply did this because trajectory of a projectile is always a concave down parabola and also because changing the equation won't have any effect on the original question. Now let's say the projectile is fired with a velocity of v at an angle Θ from horizontal.
[itex] x= v cos \theta t \\
y = v sin \theta t - gt^2 / 2 [/itex]

If you eliminate x and y from the equations, you get the eqn of trajectory which is

[itex] y = x tan \theta - \dfrac{g sec^2 \theta}{2 v^2} x^2 [/itex]

But the original trajectory is y = -x^2. Comparing the two equations

tanΘ = 0 and [itex]\dfrac{g sec^2 \theta}{2 v^2} = 1[/itex]

The radius of curvature is given by r= v^2 /g.

Solving the above eqns you can see that r indeed comes out to be 1/2!

But I'd also like to point out that this has no physical significance as Θ can never be zero in a projectile and if it is zero, then it would be wrong to call it a projectile motion. But mathematically I don't see any problem with this approach. Now, it's upto you whether you want to go with this method or not.

Cheers. :smile:
 

FAQ: Circle Inscribed in a Parabola?

What is a circle inscribed in a parabola?

A circle inscribed in a parabola is a geometric construction in which a circle is drawn inside a parabola, such that the circle touches the parabola at exactly one point and the diameter of the circle is equal to the chord of the parabola that passes through that point.

How is a circle inscribed in a parabola constructed?

To construct a circle inscribed in a parabola, the first step is to draw the parabola and identify the focus and the directrix. Then, using the focus and the directrix, draw a line perpendicular to the directrix from the focus. This line will intersect the parabola at two points. The midpoint of these two points is the center of the inscribed circle. The radius of the circle can be determined by measuring the distance from the center to the point of contact on the parabola.

What is the relationship between the circle and the parabola in this construction?

The circle and the parabola in this construction are tangent to each other at exactly one point. The point of tangency is also the point where the tangent to the parabola is perpendicular to the tangent to the circle.

What are some real-world applications of a circle inscribed in a parabola?

One real-world application of a circle inscribed in a parabola is in the design of bridges. Engineers use this construction to determine the shape and dimensions of the arches in a bridge, as it allows for a more efficient and stable design. Additionally, the construction is also used in the design of certain types of lenses, such as parabolic mirrors, used in telescopes and satellite dishes.

What are some other interesting properties of a circle inscribed in a parabola?

One interesting property of a circle inscribed in a parabola is that the tangent to the circle at the point of contact is parallel to the directrix of the parabola. Another property is that the angle between the tangent to the circle and the tangent to the parabola at the point of contact is equal to the angle between the tangent to the parabola and the axis of symmetry. Additionally, the area of the circle is always equal to the area of the region between the parabola and the circle.

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