(4a+7b)(4c-7d) = (4a-7b)(4c+7d)
(4a+7b)/(4a-7b) = (4c+7d)/(4c-7d)
Using componendo and dividendo
(4a+7b)+(4a-7b) / (4a+7b)-(4a-7b)
= (4c+7d)+(4c-7d) / (4c+7d)-(4c-7d)
Or 8a/14b = 8c/14d
Or a/b = c/d
a/(b+c) = b/(c+a) = c/(a+b)
Taking reciprocal and adding 1 to each ratio we get;
(b+c)/a + 1 = b/(c+a) + 1 = c/(a+b) + 1
Or (a+b+c)/a = (a+b+c)/b = (a+b+c)/c
So this can only be equal when a=b=c or a+b+c = 0
When a=b=c we get
a/(b+c)= ½When a+b+c = 0 we get
b+c = -a
So a/(b+c) = -1
So the ratios are ½ or -1
Let 2+?2+?2..... = t
2+?t = t
t - 2 = ?t
t = 4
Let there be 2 circles with centre O1 and OAB is the common chord
Since both passes through the center of each other as shown in figure
So O1O is the radius of both
Let O1O = r = AO1= AO
AX = AB / 2 = 5?3 cm (since OX perpendicular to chord bisects it)
AOO1 forms an equilateral triangle with on side = radius = r
Sin 60 = ?3/2 = AX / AO = 5?3/r
So r = 10 cm
So diameter = 20 cm
Mean of m observations is n
Mean of n-m observations is m
So total = nm+ (n-m)m
Total observations = n
Mean = Total / Total observations = (2mn-m^2)/n = 2m ?m^2/n
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