Two curves with two shared tangents

[KingNothing]

I'm stuck on this problem. I am hoping someone can walk me through it or get me past my choking point. The problem states:

Two equations have two shared tangent lines between them. Find the equations of these tangent lines analytically.

$$g(x)=x^2$$
$$f(x)=-x^2+6x-5$$

The first step I took was to define what is needed for two curves to share a single tangent line:

-$$g'(x_0)=f'(x_1)$$ slopes at two different x-values must be the same
-the tangent line to $$(x_0,y_0)$$ must pass through the point $$(x_1,y_1)$$ on the other curve with the same slope

I found the general equation for the tangent of a line $$f(x)$$ at $$x$$ to be $$f'(x)k-xf'(x)+f(x)$$ where $$k$$ is to be substituted for $$x$$ in the final equation.

I did this with both of the functions in the problem, and set them equal. My final result for this was $$4xk-2x^2=6k-5$$ Assuming you leave k in there and just try to get the sides looking equal, I got nowhere. After substituting x for k, there is no solution.

 

[0rthodontist]

It is easier to approach it from the perspective of the number of solutions of g(x) - L(x) = 0 and L(x) - f(x) = 0. If both equations have one solution (which since they are both quadratic means that b^2 - 4ac = 0) then L(x) is tangent to both curves. There are two solutions for m and b of L(x) = mx + b.

 

[KingNothing]

Could you go through that approach? I sort of understand your logic, and sort of don't. I don't understand what L(x) is doing when those two equations can be simplified to g(x)-f(x)=0.

 

[0rthodontist]

L(x) is the equation of the line, L(x) = mx + b for m and b that you are trying to determine.

For example this is what you do for the first equation, g(x) - L(x) = 0
x^2-(mx + b) = 0
x^2 - mx - b = 0
This equation has a unique solution iff m^2 + 4b (the expression under the radical sign in the quadratic formula) = 0. So we have m^2 + 4b = 0, and we can work with L(x) - f(x) = 0 in a similar manner to get another equation involving m and b.

The idea is that if g(x) is tangent to L(x), then g(x) - L(x) has exactly one root (only true because g(x) is quadratic, as it is here). Drawing a picture can make that clear.

You can't simply combine the equations because you are not actually stating that either g(x) - L(x) = 0 or L(x) - f(x) = 0 is true. You are just setting them equal to 0 because you need to find how many roots each has.

 

[VietDao29]

L(x) is the equation of the line, L(x) = mx + b for m and b that you are trying to determine.

For example this is what you do for the first equation, g(x) - L(x) = 0
x^2-(mx + b) = 0
x^2 - mx - b = 0
This equation has a unique solution iff m^2 + 4b (the expression under the radical sign in the quadratic formula) = 0. So we have m^2 + 4b = 0, and we can work with L(x) - f(x) = 0 in a similar manner to get another equation involving m and b.

The idea is that if g(x) is tangent to L(x), then g(x) - L(x) has exactly one root (only true because g(x) is quadratic, as it is here). Drawing a picture can make that clear.

You can't simply combine the equations because you are not actually stating that either g(x) - L(x) = 0 or L(x) - f(x) = 0 is true. You are just setting them equal to 0 because you need to find how many roots each has.

What's that for? One can differentiate f(x) with respect to x to get f'(x) = 2x, then use the tangent line equation at (x₀, y₀):
y - y₀ = f'(x₀)(x - x₀), to arrive at the equation:
y = 2x₀(x - x₀) + y₀ = 2x₀x - 2x₀² + x₀² = 2x₀x - x₀².
-----------------------
You can also approach it from this way:
Say P(x₀, y₀), P is on the graph of f(x), and Q(x₁, y₁), Q is on the graph of g(x) share the same tangent line.
So it means that:
$$\left\{ \begin{array}{l} f'(x_0) = g'(x_1) \\ \frac{y_0 - y_1}{x_0 - x_1} = f'(x_0) \end{array} \right.$$
Can you see why? Can you go from here? :)

 

[KingNothing]

No, I don't think I can. But that seems to make a lot more sense than the L(x) stuff.

I still don't see how to solve either of those methods. The first method you described is essentially what I have been trying for. The tangent equation for f(x) (the one you described was actually g(x)) ends up being $$y-y_1=f'(x_1)(x-x_1)$$ or $$y=(-2x_1+6)(x-x_1)+y_1$$ Can you check that?

Even once that's done, I still don't understand how to solve for distinct tangent lines.

 

[VietDao29]

No, I don't think I can. But that seems to make a lot more sense than the L(x) stuff.

I still don't see how to solve either of those methods. The first method you described is essentially what I have been trying for. The tangent equation for f(x) (the one you described was actually g(x)) ends up being $$y-y_1=f'(x_1)(x-x_1)$$ or $$y=(-2x_1+6)(x-x_1)+y_1$$ Can you check that?

Even once that's done, I still don't understand how to solve for distinct tangent lines.

It can be simplified to:
$$y = (-2x_1 + 6) (x - x_1) + y_1 = -2x_1 x + 2x_1 ^ 2 + 6x - 6x_1 - x_1 ^ 2 + 6x_1 - 5 = (-2x_1 + 6) x + x_1 ^ 2 - 5$$
By the way, the equation f'(x₀) = g'(x₁) means that the tangent line of f(x) at x₀, and the tangent line of g(x) at x₁ are parallel. Do you know why?
The next equation:
$$\frac{y_1 - y_0}{x_1 - x_0} = f'(x_0)$$ is to assure that Q is on the tangent line of f(x) at x₀, and P is on the tangent line of g(x) at x₁. Or in other words, PQ is the tangent line of f(x) and g(x) at x₀, and x₁ respectively.
I'll start of for you:
f'(x₀) = g'(x₁)
<=> -2x₀ + 6 = 2x₁
<=> x₀ = 3 - x₁
Now this means that the tangent line of g(x) at x₁ is parallel to the tangent line of f(x) at 3 - x₁. Do you see why?
Now you can substitute that to the equation:
$$\frac{y_1 - y_0}{x_1 - x_0} = f'(x_0)$$, and solve for x₁. What's y₁ in terms of x₁, what's y₀ in terms of x₀? Note that P, and Q are on the graph of f(x), and g(x) respectively. Can you go from here? :)

 

[0rthodontist]

What's that for?

Didn't I just explain it to KingNothing? It is a faster and simpler technique in this situation than using calculus. Again, the idea is, the following two statements are equivalent when f(x) is quadratic and L(x) is a line:
"L(x) is tangent to f(x)"
"f(x) - L(x) has exactly one root"

 

[VietDao29]

Didn't I just explain it to KingNothing? It is a faster and simpler technique in this situation than using calculus. Again, the idea is, the following two statements are equivalent when f(x) is quadratic:
"The line L(x) is tangent to the quadratic function f(x)"
"f(x) - L(x) has exactly one root"

So I'll get m² = -4b for L(x). Now I define K(x) in the same manner that's the tangent line to f(x), K(x) = lx + u. Then I'll get:
(6 + l)² = 20 - 4u. Then how can I go from there?
Am I missing something?

 

[0rthodontist]

It has to be the same line that is tangent to both quadratics. L(x) = mx+b for both quadratics. The other equation you get is 16-12m+m^2-4b = 0

 

[VietDao29]

It has to be the same line that is tangent to both quadratics. L(x) = mx+b for both quadratics. The other equation you get is 16-12m+m^2-4b = 0

Yes, it's simplier than my way. Thanks for pointing that out. :)
KingNothing, now you have 2 ways to tackle this problem. Both are good, I think. Can you go from here? :)