- #1
flemonster
- 11
- 0
Homework Statement
Given the two parabolas: [itex] f(x) = x^2 - 2x + 2[/itex] and [itex] g(x) = -x^2 - 2x - 2[/itex]. Find the equation of a line that is tangent to both curves.
Homework Equations
The given parabolas, equation for a line [itex]y = mx + b[/itex], and the derivatives of the two parabolas [itex]2x - 2[/itex] and [itex]-2x - 2[/itex]
The Attempt at a Solution
The line tangent to the two parabolas will pass through the points
[itex](x_1 , y_1)[/itex]
for the parabola [itex]f(x)[/itex] and
[itex](x_2 , y_2) [/itex]
for the parabola [itex]g(x)[/itex]
so the equations for the two lines will be,
for f' [itex]y_1 = (2x_1 - 2)x_1 +b [/itex]
and
g' [itex] y_2 = (-2x_2 - 2)x_2 + b[/itex].
Since the slopes of both lines will be the same I thought that setting the two slopes equal might get me started so I wrote
[itex]2x_1 - 2 = -2x_2 - 2 [/itex]
which gave me [itex]\frac{x_1}{x_2} = -1 [/itex].
I rearranged the two linear equations and set them equal:
[itex]y_1 - (2x_1 - 2)x_1 = y_2 - (-2x_2 - 2)x_2[/itex]
but that got me absolutely nowhere. I got the whole thing down to
[itex]x^2 _1 + x^2 _2 = \frac{y_1 - y_2}{2}[/itex]
but that doesn't help.
I know I need to limit my variables and try to get the whole thing in terms of one variable but I'm at a loss as to how to make that happen. Every time I substitute and simplify I get either one or negative one which tells me nothing. I can't figure out how relate the equations and simplify. Any help is appreciated.