Subject: Common Tangents - Calculus
Name: Anne
Who is asking: Student
Level: Secondary
Question:
I have been working on this problem for a while but I'm not sure I'm
getting the right answer:
Find the common tangents of 2y=x^{2} and 2y=-x^{2}-16
Thanks for the help. :)
Hi Anne,
Write the first equation as
f(x) = ^{1}/_{2} x^{2}
and the second as
g(x) = ^{-1}/_{2} x^{2} - 8
Suppose that P is a point on the graph of the first equation with
first coordinate a. Write the equation of the tangent line to this
graph at P. I got
y = ax - ^{1}/_{2} a^{2}
Now suppose that Q is a point on the graph of the second equation
with first coordinate b. Write the equation of the tangent line to
this graph at Q. I got
y = -bx -8 + ^{1}/_{2} b^{2}
If these two equations give the same line then the slopes are the same
a = -b
and the y-intersepts are the same
- ^{1}/_{2} a^{2} = -8
+ ^{1}/_{2} b^{2}
Solve for a and b.
Each pair (a,b) that you find gives you a pair of points P and Q. To
check you can write the equations of the tangent lines at P and Q to
verify they are identical.
Cheers,
Harley