I understand how to solve the problem but how is the answer even possible?

Integrate

Integrate

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The figure shows a point P on the parabola y=x^2 and the point Q where the perpendicular bisector of OP intersects the y-axis. As P approaches the origin along the parabola, what happens to Q? Does it have a limiting position? If so, find it.


How in the **** is the limiting position 1/2? Why does the limit not go to infinity?

INFINITY.PNG
 
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Integrate said:
The figure shows a point P on the parabola y=x^2 and the point Q where the perpendicular bisector of OP intersects the y-axis. As P approaches the origin along the parabola, what happens to Q? Does it have a limiting position? If so, find it.


How in the **** is the limiting position 1/2? Why does the limit not go to infinity?
Click to expand...
Where is point O? The origin?
 
How did you get infinity? Which expression do you get for y-coordinate of Q for arbitrary P? Do you have a drawing?
 
Integrate said:
The figure shows a point P on the parabola y=x^2 and the point Q where the perpendicular bisector of OP intersects the y-axis. As P approaches the origin along the parabola, what happens to Q? Does it have a limiting position? If so, find it.


How in the **** is the limiting position 1/2? Why does the limit not go to infinity?
Click to expand...
You're right that the answer is 1/2, and that it is surprising.

The problem is discussed here, first helping a student work it out, and then demonstrating in several ways what is happening.
 
Dr.Peterson said:
You're right that the answer is 1/2, and that it is surprising.

The problem is discussed here, first helping a student work it out, and then demonstrating in several ways what is happening.
Click to expand...
I'm reading through it now, but is this a famous problem or something?
 
I know sharing a graph isn't needed at this point but I'm practicing Desmos. See here.

Isn't there a way to upload a copy of a Desmos graph in the post?

-Dan
 
Integrate said:
This REALLY helps, but could it happen in the physical world. Say with a flag pole and yarn? It seems as if it wouldn't.
Click to expand...
Why not? How is the graph different from the "physical world"? Whatever you do with yarn (I'm not sure what you have in mind) would just be a physical version of the graph, wouldn't it?
 
I wouldn't be so incredulous if these weren't so counterintuitive. I can see why my textbook put this after the chapter review even.

Easter eggs like this really do make learning math that much more fun. I'll have to email Numberphile to make a video on this topic.


This problem needs a name.
 
What is point Q in your initial post assuming P = (k, k^2).
What happens to Q as k->0?

Why do you think the answer is so counterintuitive? If you tell why, then we will have a chance to show you why it isn't
 
Integrate said:
I wouldn't be so incredulous if these weren't so counterintuitive. I can see why my textbook put this after the chapter review even.

Easter eggs like this really do make learning math that much more fun. I'll have to email Numberphile to make a video on this topic.


This problem needs a name.
Click to expand...
I have to admit I wouldn't expect this answer of the top of my head, but, in hindsight, is it more counterintuitive than the focusing property of parabolas? :

prbl4.png
 
And since beating on a dead horse is my second name, here is an illustration of another similar limit, this time from intersections of the normals with the Y axis:

prbl3.png
 
I too didn't expect the answer to be 1/2 but I can't think of a reason why the OP thinks that the answer should have been infinity. Just visualizing or drawing the graphs you see that the limit will not be infinity--in fact it will not be a 'large' value.
 
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