1/a+1/b=1/200, (a+b)/ab=1/200, so ab/(a+b)=200,
ab=200(a+b)=200a+200b, ab-200b=b(a-200)=200a, b=200a/(a-200).
Since the equation is symmetrical in a and b (so a=200b/(b-200)), and (a,b) is an ordered pair satisfying the equation, then so is (b,a).
Also, since 1/200 is a positive number, a and b cannot both be negative, so we only need to consider positive integers for one variable. If a=b, then 2/a=2/b=200, so a=b=400 (ordered pair (400,400)). Therefore we only need to find integers between a=1 and 400 that satisfy integer b=200a/(a-200), because these will generate values for b (some of which will be negative). Having found all the (a,b) set, we automatically have another set formed of (b,a).
200=2352, so we have 12 factors: 2, 4, 5, 8, 10, 20, 25, 40, 50, 100, 200 and 1.
Considering b=200a/(a-200), a-200 has to be an integer (positive or negative). Since b is also an integer a-200 must divide into 200 for some values of a, making a-200 one of the factors of 200. Examples: a-200=2, a=202 then b=20200; a-200=1, a=201, b=40200; a=204, b=10200; a=205, b=8200; etc. up to a=300, b=600. Take (a,b)=(205,8200): 1/205+1/8200=1/200 so a random check proves the original equation.
Since there are 12 factors, there must be 12 pairs of values for positive a and b:
(a,b)=(201,40200), (202,20200), (204,10200), (205,8200), (208,5200), (210,4200), (220,2200), (225,1800), (240,1200), (250,1000), (300,600), (400,400).
We must also consider negative factors: a-200=-1, a=199, b=-39800, etc.:
(199,-39800), (198,-19800), (196,-9800), (195,-7800), (192,-4800), (190,-3800), (180,-1800), (175,-1400), (160,-800), (150,-600), (100,-200).
So far there are 12+11=23 integer pairs. We can also reverse the order to get a further 23, making 46 ordered pairs.
We need to discover any values of a we might have missed.
The total set of (a,b) where a is positive is:
(40,-50), (75,-120), (100,-200), (120,-300), (136,-425), (150,-600), (160,-800), (168,-1050), (175,-1400), (180,-1800), (184,-2300), (190,-3800), (192,-4800), (195,-7800), (196,-9800), (198,-19800), (199,-39800), (201,40200), (202,20200), (204,10200), (205,8200), (208,5200), (210,4200), (216,2700), (220,2200), (225,1800), (232,1450), (240,1200), (250,1000), (264,825), (280,700), (300,600), (325,520), (360,450), (400,400). The highlighted pairs are an additional 12 integer pairs where a-200 divides into 200a.
By reversing these (except for (400,400)) we have a further 34 ((450,360), (520, 325), etc.) making 35+34=69 ordered pairs.