Here is how the problem reads, the graphing part I got, it's just the problem itself.

Graph the two parabolas y=x^2 and y=-x^2 +2x -5 in the same coordinate plane. Find the equations of the two lines simultaneously tangent to both parabolas.

I got as far as just finding the derivatives which as 2x and -2x+2, but I can't the equations.

2. Originally Posted by doctorgk
Here is how the problem reads, the graphing part I got, it's just the problem itself.

Graph the two parabolas y=x^2 and y=-x^2 +2x -5 in the same coordinate plane. Find the equations of the two lines simultaneously tangent to both parabolas.

I got as far as just finding the derivatives which as 2x and -2x+2, but I can't the equations.
Given an parabola $y=ax^2+bx+c,a\not = 0$ and a line $y=px+q$. This line is tangent to the parabola if and only if $(b-p)^2 - 4a(c-q)=0$. So if $px+q$ is tangent to $x^2$ then it means $p^2 + 4q = 0$ and it tangent to $-x^2+2x-5$ that means $(p-2)^2 - 4(q+5)=0$. The solutions to this system of equations is $(p,q) = \{(-2,-1),(4,-4)\}$.

3. ## Okay but still a bit lost

how did you get that formula? is there a way to solve this with derivatives?

4. Originally Posted by ThePerfectHacker
Given an parabola $y=ax^2+bx+c,a\not = 0$ and a line $y=px+q$. This line is tangent to the parabola if and only if $(b-p)^2 - 4a(c-q)=0$. So if $px+q$ is tangent to $x^2$ then it means $p^2 + 4q = 0$ and it tangent to $-x^2+2x-5$ that means $(p-2)^2 - 4(q+5)=0$. The solutions to this system of equations is $(p,q) = \{(-2,1),(4,-4)\}$.
the other solution to the system is (-2,-1), not (-2,1)

5. Originally Posted by doctorgk
how did you get that formula? is there a way to solve this with derivatives?
Given a parabola $y=ax^2+bx+c,a\not = 0$ and $y=px+q$ if they intersect exactly once then the equation (of the intersection) $ax^2+bx+c = px+q$ has exactly one solution, this is a quadradic so, $ax^2+(b-p)x+(c-q)=0$. We want this quadradic to has exactly one real solution that happens when the discrimant is zero.

(We want it to have exactly one solution because a tangent line to a parabola intersects it exactly one time. )

the other solution to the system is (-2,-1), not (-2,1)
Okay I fix it.

6. ## Okay

makes sense now, thank you very much