Question

Can somebody help me ??

make a sketch showing that there are two straight lines, each of which is tangent to both of the parabolas y=x^{2}+4x+1 and y=-x^{2}+4x-1. Find equations of the two lines.

Answer 1

Drawing it is hard, I suggest you use something like the ode java grapher to get a quick graph of the functions. ( http://dlippman.imathas.com/g1/fullgrapher2d.html)

That aside, the two lines are both tangent to both parabolas, so if you differentiate both, to get:

f'1(x) = 2x +4 and f'2(x) = -2x +4

You then set these equal to each other:

2x + 4 = -2x + 4

to get a value of x, x = 0, which you can then plug back into either derivative to get the value of the derivative at that point.

Now use 0 in each of your original functions, and hey presto, you've got two coordinates, and the m value for a line of the form:

y=mx+c.

Interestingly, the lines are the same line if you solve it through.

Answer 2

Mind, the last bracket at the link should not have been included. The link is
http://dlippman.imathas.com/g1/fullgrapher2d.html