Consider the following for the next items that follow:

A parabola passes through (1, 2) and satisfies the differential equation $\frac{d y}{d x}=\frac{2 y}{x}, x>0, y>0 .$

What is the directrix of the parabola ?

Consider the following for the next items that follow:

A parabola passes through (1, 2) and satisfies the differential equation $\frac{d y}{d x}=\frac{2 y}{x}, x>0, y>0 .$

What is the length of latus rectum of the parabola?

Consider the following for the next items that follow:

Let $f(x)=\frac{a^{x-1}+b^{x-1}}{2}$ and g(x) = x - 1.

What is $\displaystyle \lim _{x \rightarrow 1} \frac{f(x)-1}{g(x)}$ equal to?

Consider the following for the next items that follow:

Let $f(x)=\frac{a^{x-1}+b^{x-1}}{2}$ and g(x) = x - 1.

What is $\displaystyle \lim _{x \rightarrow 1} f(x)^{\frac{1}{g(x)}}$ equal to?