In 2009, there were 26 finale questions in Tianjin senior high school entrance examination: (2) Under the condition of (1), if the two intersections of the function Y 1 and the image of Y2 are A and B, when the area of the triangle ABM is112 cubic, find the value of t;

Note: The coordinate given by M in the question is (t, t).

Question: Why |t-T|=√2h (1)?

The formula (1) can be interpreted in two ways:

One method directly uses the distance formula from point to straight line;

The distance from the point (x 1, y 1) to the straight line ax+by+c=0 is:

D=|ax 1+by 1+c|/ under the quadratic root sign (a 2+b 2)

Therefore, the distance from the point M( t, t) to the straight line y=x, that is, x-y=0 is:

H=d=|t-T|/ Under the square root (12+12) = | t-t |/√ 2.

Therefore |t-T|=√2h (1).

The second method:

Let the straight line passing through point M( t, t) and perpendicular to the straight line Y=X be Y=kX+b and the vertical foot be n.

Then k=- 1

T=-t+b

There is b=T+t

Therefore, a straight line passing through point M( t, t) and perpendicular to line Y=X is y =-x+t+t.

Simultaneous equation: Y = X；;

Y=-X+T+t

X=(T+t)/2。

Y=(T+t)/2

The vertical foot is N((T+t)/2, (T+t)/2).

According to the distance formula between two points

H = Mn = | t-t |/√ 2。

Therefore |t-T|=√2h (1).