Calculus Problem That I Need Help With

[timmyyan]

Here is the problem:

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

Any help would be appreciated. Thank you.

 

[Dr.Peterson]

Here is the problem:

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

Any help would be appreciated. Thank you.

My first suggestion is to make the graph as they tell you to do, and show it to us; that can at least help you visualize the task.

Next, please tell us what tools you have available, so we can know what methods to suggest. Are you doing single-variable calculus, or can you handle things like partial derivatives, in case that should be useful? What is the most recent topic you learned, assuming this is for a class? Also, have you tried writing several equations that must be true to meet the stated conditions?

The more you can show of your current knowledge and what you have tried, the more we can help.

 

[timmyyan]

Thank you! Here is how I started:

I guess what I am confused about is:
A) are those the correct values for C?
B) I think I have to plug those values back into the derivative of C and then create the equations (maybe in point-slope form)?

Thanks.

 

[BigBeachBanana]

This problem can be solved without calculus by taking advantage of our knowledge of quadratics and discriminants.
A line [imath]y=mx+k[/imath] is tangent to a quadratic iff [imath]mx+k=ax^2+bx+c[/imath]
[math]\begin{cases} mx+k=x^2 \\ mx+k=-x^2+2x-5 \end{cases} \Longleftrightarrow \begin{cases} x^2-mx-k=0 \\ x^2+(-2+m)x+5+k=0 \end{cases}[/math]Now, set their discriminants equal to 0, and you'll have a system of two equations. Solve for [imath]m,k.[/imath]

 

[Dr.Peterson]

Thank you! Here is how I started:
View attachment 30830View attachment 30831
I guess what I am confused about is:
A) are those the correct values for C?
B) I think I have to plug those values back into the derivative of C and then create the equations (maybe in point-slope form)?

Thanks.

First, your graph is inaccurate, though that doesn't affect your work.

Second, check your algebra:


I don't notice any other errors; try fixing that, and see if it works better.

Once you have all the coordinates of the points, just find the equation of the line through them.

Then make a better graph and see if your lines look right.

 

[timmyyan]

Thank you so much for your help! Makes sense now