Determine the parameter a so that the parabola y=ax²

Determine the parameter $a$ so that the parabola $y=ax^{2}$ touches the curve $y=\ln x$. Find the coordinates of the point of tangency.

Step 1: The two tangency conditions

\[ ax_{0}^{2}=\ln x_{0}, \qquad 2ax_{0}=\frac1{x_{0}} \;\Longrightarrow\; a=\frac{1}{2x_{0}^{2}}. \tag{1,2} \]

Step 2: Solve for $x_0$

\[ \frac{1}{2x_0^2}\cdot x_0^2 = \ln x_0 \;\Longrightarrow\; \frac12=\ln x_0 \;\Longrightarrow\; x_0 = \sqrt e. \]

Step 3: Verification

\[\frac{1}{2e}\cdot e = \frac12 = \ln\sqrt e\ \checkmark; \qquad 2\cdot\frac{1}{2e}\cdot\sqrt e = \frac{1}{\sqrt e}\ \checkmark\]