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# Tangents to cubics

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1. I came the following difficult question.
The curve C with equation y=x3+3x2-3x is reflected in the line y=x onto the curve C'.
(a) Find the equation of the curve C'.
(b) Find the equation of the line L, which is tangent to both curve C and curve C'.
[Your equation for L should be in its exact form. No credit will be given for any solutions found using differentiation.]
This is actually a difficult question. Any ideas?
Diagram
2. (Original post by Ano123)
I came the following difficult question.
The curve C with equation y=x3+3x2-3x is reflected in the line y=x onto the curve C'.
(a) Find the equation of the curve C'.
(b) Find the equation of the line L, which is tangent to both curve C and curve C'.
[Your equation for L should be in its exact form. No credit will be given for any solutions found using differentiation.]
This is actually a difficult question. Any ideas?
Wth why would they ban differentiation what's the point haha. What level is that question from? Also Is there a diagram for it?
3. (Original post by Nayzar)
Wth why would they ban differentiation what's the point haha. What level is that question from? Also Is there a diagram for it?
It's FP1, there is a diagram. But I haven't taken a picture of the question. I'll upload one.
4. (Original post by Ano123)
This is actually a difficult question. Any ideas?
I think that the biggest hint that I can give you is that the question is much easier than you seem to think it is. Another hint in the spoiler:
Spoiler:
Show
Think about the relationship between the line L and the line y = x.
5. (Original post by Gregorius)
I think that the biggest hint that I can give you is that the question is much easier than you seem to think it is. Another hint in the spoiler:
Spoiler:
Show
Think about the relationship between the line L and the line y = x.
It has gradient of -1 right? I know that and have seen questions like this before. But why does it have to be -1? What's an intuitive way of understanding this?
6. (Original post by Ano123)
What's an intuitive way of understanding this?
Look for symmetries.
7. Does anyone have an answer?
8. (Original post by Ano123)
I could be missing a trick but the answer isn't at all obvious to me.

If we zoom out a bit, then we can see there are two lines meeting the question's criterion.

And the algebra was a bit tortourous (and I got it wrong!); looking for a repeated root in the equation -x+c =x3+3x2-3x

And if we change the equations slightly, by adding 3 say, then there are even more solutions and the additional ones are not perpendicular to the line of symmetry:

Attachment 545941545943
Attached Images

9. (Original post by Ano123)
Sorry for delay; got called away. It's clear from symmetry that the slope of the line is -1. So the equation of the line can be expressed as . We need to find . This line is tangent to the original curve, so we need to find such that has a double root at some value of . (It is tangent there and intersects the curve for some smaller value of y not plotted on the original graph). So, solve the equation for the value of . You should find that the same equation that calculus will give drops out!
10. (Original post by ghostwalker)
I could be missing a trick but the answer isn't at all obvious to me.

If we zoom out a bit, then we can see there are two lines meeting the question's criterion.

PRSOM. I assumed that the picture given was part of the question and therefore the line shown was what was required. Then the algebra reduces to finding a double root (with the line intercepting the original curve, as you have shown, for a more negative value of y).
11. Thank you for your help. I did get the answer but I used the discriminant and set it equal to 0 for the intersection of y=c-x and found c by solving the quadratic formed in c. As ghostwalker said there are actually 2 lines, but L is indicate don the diagram. The repeated root method is a nice one as well but I didn't think to do this.
There's nothing you lot can't do it seems.

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