Let $C$ be the parabola $y = x ^ { 2 } + 1$ with origin O, and let P be the point $(a, 2a)$.

(1) Find the equation of the line passing through point P and tangent to parabola $C$.

The equation of the tangent line at point $(t, t ^ { 2 } + 1)$ on $C$ is

$$y = \square t x - t ^ { 2 } + 1$$

If this line passes through P, then $t$ satisfies the equation

$$t ^ { 2 } - \square a t + \text { エ } a - \text { オ } = 0$$

so $t = \square a -$ カ y. Therefore

When $a \neq$ ケ, there are 2 tangent lines to $C$ passing through P, and their equations are

$$y = ( \square a - \square ) x - \text { シ } a ^ { 2 } + \text { ス } a$$

a ⋯⋯⋯(1)

and

$$y = \text { せ } x$$

(2) Let $\ell$ be the line represented by equation (1) in (1). If the intersection of $\ell$ and the $y$-axis is $\mathrm { R } ( 0 , r )$, then $r = -$ シ $a ^ { 2 } +$ ス $a$. For $r > 0$,

ソ $< a <$ タ

タ and in this case, the area $S$ of triangle OPR is

$$S = \text { チ } \left( a ^ { \text {ツ } } - a \text { 园 } \right)$$

When ソ $< a <$ タ , examining the increase and decrease of $S$ , we find that $S$ attains its maximum value at $a = \frac { \text { ト } } { \text { ナ } }$

(3) When □ソ $< a <$ □タ, the area $T$ of the region enclosed by the parabola $C$, the line $\ell$ in (2), and the two lines $x = 0, x = a$ is

$$T = \frac { 1 } { \square } \text { ハ } a ^ { 3 } - \square a ^ { 2 } + \square$$

ト

In the range $\frac { 1 } { \text { ナ } } \leqq a <$ タ, $T$ is □ヘ.

In the range, $T$ is □ヘ. Choose the correct answer for □ヘ from the following options (0) through (5).

(0) decreasing
(1) attains a local minimum but not a local maximum
(2) increasing
(3) attains a local maximum but not a local minimum
(4) constant
(5) attains both a local minimum and a local maximum