Find tangent lines to both curves

[drawar]

Homework Statement

Find the equation of all straight lines, if any, that are tangent to both the curves [itex]y = {x^2} + 4x + 1[/itex] and [itex]y = - {x^2} + 4x - 1[/itex].

Homework Equations

The Attempt at a Solution

Suppose such a line exists and its slope is m. Let [itex]({x_1},{y_1})[/itex] and [itex]({x_2},{y_2})[/itex] be the tangent points on the curves [itex]y = {x^2} + 4x + 1[/itex] and [itex]y = - {x^2} + 4x - 1[/itex] respectively.
Then [itex]{y_1} = {x_1}^2 + 4{x_1} + 1[/itex] and [itex]{y_2} = - {x_2}^2 + 4{x_2} - 1[/itex].
The slope of the curves at [itex]{x_1}[/itex] is [itex]2{x_1} + 4[/itex] and at [itex]{x_2}[/itex] is [itex]-2{x_2} + 4[/itex]. Thus [itex]m=2{x_1} + 4=-2{x_2} + 4 \Rightarrow {x_1} = - {x_2}[/itex]. Moreover, [itex]m = \frac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}} = \frac{{( - {x_2}^2 + 4{x_2} - 1) - ({x_1}^2 + 4{x_1} + 1)}}{{{x_2} - {x_1}}} = \frac{{{y_1}}}{{{x_1}}}[/itex].
What should be the next steps here? Thanks!

 

[SammyS]

Homework Statement

Find the equation of all straight lines, if any, that are tangent to both the curves [itex]y = {x^2} + 4x + 1[/itex] and [itex]y = - {x^2} + 4x - 1[/itex].

Homework Equations

The Attempt at a Solution

Suppose such a line exists and its slope is m. Let [itex]({x_1},{y_1})[/itex] and [itex]({x_2},{y_2})[/itex] be the tangent points on the curves [itex]y = {x^2} + 4x + 1[/itex] and [itex]y = - {x^2} + 4x - 1[/itex] respectively.
Then [itex]{y_1} = {x_1}^2 + 4{x_1} + 1[/itex] and [itex]{y_2} = - {x_2}^2 + 4{x_2} - 1[/itex].
The slope of the curves at [itex]{x_1}[/itex] is [itex]2{x_1} + 4[/itex] and at [itex]{x_2}[/itex] is [itex]-2{x_2} + 4[/itex]. Thus [itex]m=2{x_1} + 4=-2{x_2} + 4 \Rightarrow {x_1} = - {x_2}[/itex]. Moreover, [itex]m = \frac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}} = \frac{{( - {x_2}^2 + 4{x_2} - 1) - ({x_1}^2 + 4{x_1} + 1)}}{{{x_2} - {x_1}}} = \frac{{{y_1}}}{{{x_1}}}[/itex].
What should be the next steps here? Thanks!

You have it nearly done.

Well, you have:

[itex]x_1=-x_2[/itex]

[itex]m=-2{x_2} + 4[/itex]

[itex]\displaystyle m=\frac{{( - {x_2}^2 + 4{x_2} - 1) - ({x_1}^2 + 4{x_1} + 1)}}{{{x_2} - {x_1}}}[/itex]​

Plug -x₂ in for x₁ in that last equation & equate that to your other expression for m that has only x₂ in it.

Solve for x₂.

 

[drawar]

Gosh I am so blind, thank you so much!