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hmm_marie

Member

Registered: 2006-10-09

Posts: 7

tangent of two parabolas, how do i solve this problem?

A parabola with the equation y= ax² + bx + c, a>0 touches the two parabolas p1 and p2 with the equations y= -x²+(b1)x + (c1) and y= -x² + (b2)x + (c2) at the points A and B. One has to show that the common tangent of the parabolas p1 and p2 is parallel to the straight line AB.

I really don't get this one... could anybody help me please?

fgarb

Member

Registered: 2006-03-03

Posts: 89

Re: tangent of two parabolas, how do i solve this problem?

This looks like a quite tedious problem. Sorry, I don't have time to go through and do it myself, but I'll try and give you some advice that might help if you're struggling to get started:

The first thing to do is draw a picture. From looking at the signs of the x^2 terms, you know that the first parabola opens up and the other two parabolas open down. If you draw some examples, you will see that two parabolas opening in oppositte directions can touch each other in 0, 1, or 2 places. From the way your problem is worded, I think it's saying that the up facing parabola touches each of the other ones in one place - otherwise there would be more than just points A and B. So draw your picture with The up facing parabola between the two down facing ones so that it just grazes them as it passes by (remember that the down facing parabolas probably don't start from the same height). Label the touching points A and B and draw a line between them. Now draw another line that is tangent to both of the down facing parabolas - there is only one way to do this. Your job is to show that these two lines are parallel to each other, meaning that they have the same slope.

So find the slopes, and show they are equal. To find the slope of the line tangent to the parabolas, first find the slope of a line tangent to each of the parabolas at some point along them. This is easy if you know calculus, but more complicated otherwise. Then you need to compare your results to find points at which the slope is the same for both parabolas - these are your tangent points, and take that slope as your answer.

Now you need to show that this slope is the same as the slope of the line joining A and B. First you have to find where points A and B are. To do this, simply set the equations for the intersecting parabolas equal to each other to find the point of intersection. Remember that the intersection only occurs in one place, so the stuff under the square root when you solve the quadratic equation must vanish. Once you have points A and B, find the slope of the line that joins them, and show it is equal to the slope of the tangent line - then you're done!

Tough problem!

hmm_marie

Member

Registered: 2006-10-09

Posts: 7

Re: tangent of two parabolas, how do i solve this problem?

hey thank you very much... i got nearly that far aswell. my problem there was that i didn't know how to get a tangent for BOTH parabolas facing downwards... so when i took the derrivative it was -2x + b1 and for the other one -2x + b2. do they have to be set as egual to get the COMMON tangent? if so, b1 and b2 have to be the same... and in that case they would have to be above each other?? oh my gosh i am sorry, but i dont get it...

fgarb

Member

Registered: 2006-03-03

Posts: 89

Re: tangent of two parabolas, how do i solve this problem?

Wow, that threw me for a bit there (kind of embarresed actually )

You have to be careful what you mean by x here. When you find the slope of the tangent line by taking the derivative, you need to remember which parabola you're talking about. In your case, you required the tangents for both parabolas to have identical slopes at the same point x. As you pointed out, this can only happen if the parabolas are centered over one another, which isn't the setup of the problem. What the problem is really saying is that you're looking at two distinct tangent points - one on each parabola, which are connected by the same line.

So, if the tangent on the first parabola happens at point x1, by differentiation you know m = -2x1 + b1, and similarly the point on the second parabola gives m = -2x2 + b2. These equations give you the x-values for the two tangent points on your line, so all you have to do is plug them into your parabola equations to get the y-values and you're done. Feel free to ask if you have more questions.

(edited for clarity)
(again)

Last edited by fgarb (2006-10-10 16:28:26)

John E. Franklin

Member

Registered: 2005-08-29

Posts: 3,588

Re: tangent of two parabolas, how do i solve this problem?

The b might move the parabola graph down and leftward by a value we should research.
The a makes the parabola narrower.
The c moves the parabola up.

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igloo myrtilles fourmis

fgarb

Member

Registered: 2006-03-03

Posts: 89

Re: tangent of two parabolas, how do i solve this problem?

That's true for the up facing one. For the down facing parabolas like y=-x²+(b1)x + (c1), you can complete the square to show:

y = -(x + (b/2))^2 + c + (1/4)b^2

This shows the center is x = -b/2 and the maximum is c + 1/4b^2

So increasing b moves your parabola left and up, and increasing c moves the parabola up. You might be able to find another way to solve the problem using these properties, but you should also be able to solve the problem just by showing the slopes of the lines are equal from here.

Edit: Blargh. No, completing the square is y = -(x - (b/2))^2 + c + (1/4)b^2, meaning center is +b/2, so it moves right and up. Time for me to go to bed.

Last edited by fgarb (2006-10-10 17:23:41)