Find m so equations have common tangent

Question

There are given two equations:

$f$: $y=\ln x$ and $g$: $y=mx^2$

The question is:

Find values of $m$ so these equations have common tangent.

Solution

I found the derivative of $f(x)$ and $g(x)$ so I got

$1/x = 2mx$.

$m = 1/2x^2$

Answer 1

A point on the curve $f$ has coordinates $(t,\ln t)$. The tangent line to $f$ at this point has equation $$ y-\ln t=\frac{1}{t}(x-t) $$ that can be rewritten $$ y=\frac{x}{t}-1+\ln t $$

The tangent line to a point $(u,mu^2)$ to the curve $g$ has equation $$ y-mu^2=2mu(x-u) $$ that is $$ y=2mux-mu^2 $$ The two tangent coincide if and only if the system $$ \begin{cases} \dfrac{1}{t}=2mu \\[6px] -1+\ln t=-mu^2 \end{cases} $$ has solution (in the unknowns $t$ and $u$).