Sketch the graphs of the two equations y=x^2 and y= -x^2+6x-5, and sketch the two lines that are tangent to both graphs. Find the equations of these lines.
Actually there are only 2 unknowns
the tangent lines at (a,f(a)) and (b,f(b) respectively are:
1)y= 2a (x-a) + a^2
2) y= (-2b+6)(x-b) - b^2 + 6b -5
The slopes are the same so 2a = -2b + 6
or a= -b+3
Also if the lines tangent at (a,f(a)) and (b,f(b) are the same then 1) must intersect 2) at (b,f(b)
So -b^2 + 6b - 5 = 2a (b-a) + a^2
so we have the system:
3) a= -b+3
4) -b^2 + 6b - 5 = 2a (b-a) + a^2
using 3) in 4)
-b^2 + 6b -5 = 2(-b+3)(b- (-b+3)) + (-b+3)^2
expand and solve
See attachment