Finding a connection track to two railroads (circle and parabola)

[dannyxyeah]

There are two railroads, represented by the curves y=(1/2)x²+7, and x²+y²=1. I am supposed to find a straight line connection so that a atrain can get from one curve to the other.

I have drawn a simple diagram to help me picture things:
http://i50.tinypic.com/2saeccx.png
where (a,b) and (c,d) will be the points at which the line will connect with each curve.

I've found the derivatives to the curves to be y'=x and y'=-x/y respectively. I understand that these give me the slopes of the tangent lines for each curve, so would I be correct in reasoning that x=-x/y since the slopes should be the same?

 

[Mark44]

There are two railroads, represented by the curves y=(1/2)x²+7, and x²+y²=1. I am supposed to find a straight line connection so that a atrain can get from one curve to the other.

I have drawn a simple diagram to help me picture things:
http://i50.tinypic.com/2saeccx.png
where (a,b) and (c,d) will be the points at which the line will connect with each curve.

I've found the derivatives to the curves to be y'=x and y=-x/y respectively.

Probably a typo, but the 2nd one should be y' = -x/y.

I understand that these give me the slopes of the tangent lines for each curve, so would I be correct in reasoning that x=-x/y since the slopes should be the same?

Yes.

Note that there is another path that works, from the left side of the circle to the right side of the parabola.

 

[dannyxyeah]

Probably a typo, but the 2nd one should be y' = -x/y.

Yes, my mistake!

Yes.

Note that there is another path that works, from the left side of the circle to the right side of the parabola.

Right, that makes sense. Thanks! How do you reckon I should proceed from here? My mind is a total blank.

 

[Mark44]

Actually, I think this is a bit more complicated than setting the derivatives fo the two curves equal. You also want the line between (a, b) on the parabola and (c, d) on the circle to have the same slope as both derivatives you found.

The problem is that you could have two tangents on the two curves with the same slope, but they aren't necessarily connected, if you follow what I'm saying.

 

[dannyxyeah]

That's exactly why I'm having trouble working out how to make sure that the tangent line passes through (a,b) and (c,d) :/

 

[SammyS]

That's exactly why I'm having trouble working out how to make sure that the tangent line passes through (a,b) and (c,d) :/

What's the equation of a line tangent to y = (1/2)x² + 7 and passing through point (a, b) which is on the parabola?

What's the equation of a line tangent to x² + y² = 1 and passing through point (c, d) which is on the circle?

 

[dannyxyeah]

What's the equation of a line tangent to y = (1/2)x² + 7 and passing through point (a, b) which is on the parabola?

y'=x
m=a
y-y1=m(x-x1)
y-b=(a)(x-a)

What's the equation of a line tangent to x² + y² = 1 and passing through point (c, d) which is on the circle?

y'=-x/y
m=-a/b
y-y1=m(x-x1)
y-b=(-a/b)(x-a)

Is this what you mean?

 

[SammyS]

y'=x
m=a
y-y1=m(x-x1)
y-b=(a)(x-a)

That's good for the line tangent to the parabola at (a, b). But you can additionally express b in terms of a, since b = (1/2)a² + 7 .

y'=-x/y
m=-a/b
y-y1=m(x-x1)
y-b=(-a/b)(x-a)

Is this what you mean?

For the circle, x + y² = 1, you should use a different point, such as (c, d).

Then y-d = (-c/d)(x-c) .

Furthermore, as you already mentioned above, the slopes of the two lines must be equal, so now given a point, (c, d) on the unit circle, you can find the value of a, and plug that into the equation of the line tangent to the parabola.

 

[dannyxyeah]

Furthermore, as you already mentioned above, the slopes of the two lines must be equal, so now given a point, (c, d) on the unit circle, you can find the value of a, and plug that into the equation of the line tangent to the parabola.

I don't really follow this last part; how do I know which point on the circle (or likewise, on the parabola) I am supposed to choose?

 

[SammyS]

I don't really follow this last part; how do I know which point on the circle (or likewise, on the parabola) I am supposed to choose?

What is the slope of the line y-b = (a)(x-a) ?

What is the slope of the line y-d = (-c/d)(x-c) ?

Equate those slopes.

 

[dannyxyeah]

What is the slope of the line y-b = (a)(x-a) ?

What is the slope of the line y-d = (-c/d)(x-c) ?

Equate those slopes.

a=(-c/d), but I've had that since the original post, no?

 

[SammyS]

a=(-c/d), but I've had that since the original post, no?

Beyond that, both lines must have the same y-intercept. Right?

There is some (maybe a lot of) algebra to do.

 

[SteamKing]

There are actually four possible connections. You can have a and c both positive or both negative, and two possibilities where a is negative and c is positive or vice versa.

 

[dannyxyeah]

I've mostly lost motivation in this problem. I don't understand the meaning of what I'm doing. :(

 

[Mark44]

I've mostly lost motivation in this problem. I don't understand the meaning of what I'm doing. :(

You're trying to find a line that:
1. Joins the parabola and the circle, and
2. Is tangent to both the parabola and the circle.

 

[SammyS]

I've mostly lost motivation in this problem. I don't understand the meaning of what I'm doing. :(

Unfortunately, in this new format for Physics Forums, there is no button to press which can provide the necessary motivation.

Solve the following for y, and try to simplify the right hand side.

\displaystyle y-d= \frac{-c}{d}(x-c)

See if that leads you anywhere. Whether it does or not, please give your result.