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).