In 2009, the last question in the middle school entrance examination of mathematics in Hangzhou.

Solution: (1) Let point B (m, n) be in the first quadrant, then tan∠POB=n/m= 1/9, m=

And b on the inverse proportional function image to obtain mn.

The solution is m=3 (negative value discarding) and point B (3, 1/3).

∫AB∑x axis, A( 1/3, 1/3), AB

8/3

(2) According to the meaning of the question, the parabolic opening is downward, and let A(a, a), B( 1/a, a), ab.

1/a

8/3

∴3a^2

three

0, get one.

-3 or a

1/3,

When a man

At -3, points A(-3, -3) and B(- 1/3, -3).

Vertex on y axis =

On x, ∴ vertex is (-5/3, -5/3).

Let the quadratic function be y

k(x+5/3)^2

5/3, substitute it into point A to get k=-3/4.

Then the quadratic function is y.

-3/4(x+5/3)^2-5/3

Similarly, when a person

At-1/3, and the resolution function is y.

-3/4(x-5/3)^2+5/3

(3) let A(a, a), B( 1/a, a), and the axis of symmetry of the parabola is x.

a/2

1/2a

Let the quadratic resolution function be y.

9/5(x-2)[x-(a+ 1/a)+2], substitute a to get a.

3 or a=6/ 13

The distance from point P to straight line AB is 3 or 6/ 13.