find tangent equation ( y=x^2 and y=x^2-4x+5) have common tangent.
1. Choose an arbitrary point on the first parabola: $\displaystyle T(t, t^2)$
2. Calculate the equation of the tangent in T at the first parabola:
$\displaystyle y-t^2=2t(x-t)~\implies~y=2tx-t^2$
3. Calculate the point of intersection ofthe tangent and the second parabola:
$\displaystyle 2tx-t^2 = x^2-4x+5~\implies~x = (2+t)\pm\sqrt{4+4t+t^2-5-t^2}$
4. The tangent of the first parabola is tangent to the second parabola too if the radical equals zero:
$\displaystyle 4t-1=0~\implies~t=\dfrac14$
5. Therefore the equation of the common tangent is:
$\displaystyle y = \frac12 x-\frac1{16}$