Practice problems/A challenging tangent line problem

For the following questions, consider the function

$$y = f(x) = (x+2)^2.$$

1. Find the equations of all lines tangent to $y=f(x)$ that pass through the point $(1,1)$. Note that this point is not on the graph of $f$.
2. Is there a point $Q=(x_{1},y_{1})$ on the line $y=x$ such that there is ONLY one tangent line to the curve $y=f(x)$ passing through Q?

Justify your answer.