As shown in the figure, it is known that there is a straight line and a curve in the rectangular coordinate system. This straight line, X axis and Y axis intersect at point a and point B respectively, and OA = ob = 1 This curve is a branch of the image of the function y = 2x / 1 in the first quadrant. Point P is any point on this curve, and its coordinates are (a, b). The perpendicular lines PM and PN made from point P to X and Y axes, and the perpendicular feet are m and n. the straight line AB intersects PM and PN at points E and f respectively. (1) it is proved that AF # 8226; be = 1 (2) the coordinates of points E and F are obtained respectively (the coordinates of point e are represented by the algebraic expression of a, and the coordinates of point e are expressed by the algebraic expression of A, Using the algebraic expression of B to express the coordinates of point F, we only need to write the results, not the calculation process); (3) Calculate the area of △ OEF (the result is expressed by the algebraic formula containing a and b); (4) Please prove whether △ AOF and △ BOE are necessarily similar; if not, briefly explain the reasons (5) When the point P moves on the curve y = 2x / 1, △ OEF changes accordingly. Point out the size of the angle whose size remains unchanged among the three internal angles of △ OEF, and prove your conclusion

As shown in the figure, it is known that there is a straight line and a curve in the rectangular coordinate system. This straight line, X axis and Y axis intersect at point a and point B respectively, and OA = ob = 1 This curve is a branch of the image of the function y = 2x / 1 in the first quadrant. Point P is any point on this curve, and its coordinates are (a, b). The perpendicular lines PM and PN made from point P to X and Y axes, and the perpendicular feet are m and n. the straight line AB intersects PM and PN at points E and f respectively. (1) it is proved that AF # 8226; be = 1 (2) the coordinates of points E and F are obtained respectively (the coordinates of point e are represented by the algebraic expression of a, and the coordinates of point e are expressed by the algebraic expression of A, Using the algebraic expression of B to express the coordinates of point F, we only need to write the results, not the calculation process); (3) Calculate the area of △ OEF (the result is expressed by the algebraic formula containing a and b); (4) Please prove whether △ AOF and △ BOE are necessarily similar; if not, briefly explain the reasons (5) When the point P moves on the curve y = 2x / 1, △ OEF changes accordingly. Point out the size of the angle whose size remains unchanged among the three internal angles of △ OEF, and prove your conclusion

(1) The intercept of E (a, 1-A), f (1-B, b) (2) line on X axis and Y axis is 1. Its equation is x + y = 1, x + Y - 1 = 0o and its distance is h = | 0 + 0 - 1 | / √ (1 + 1) = 1 / √ 2ef = √ [(a - 1 + b) & # 178; + (1 - A - b) & # 178;] = (√ 2) | a + B - 1 | this curve is a function

Scale______ Kilogram items can make the pointer rotate 90 ° clockwise

As shown in the picture, the pointer can be rotated 90 ° clockwise by placing 2kg objects

The straight line L, x-2y + 2 = 0, rotates 45 degrees counterclockwise around the point P (- 2,0), and finds his linear equation

The slope of the line x-2y + 2 = 0 is k = - A / b = - 1 / (- 2) = 1 / 2
That is k = TGA = 1 / 2
The slope becomes K1 = TG (a + 45) = (TGA + Tg45) / (1-tga * Tg45)
K1=(1/2+1)/(1/2)=3
So the slope of the straight line is 3, and it passes through the point P (- 2,0)
Let the linear equation be y = K1X + B
y=3x+b
Passing point P (- 2,0)
0=3*-2+b
b=6
So the equation of the straight line is y = 3x + 6 (or 3x-y + 6 = 0)