Question

if a : b :: c: d, prove that, (a - b) : (a + b) :: (c - d) : (c + d) also if a/b = c/d, prove that a^2 + c^2/b^2 + d^2 = ac/bd please show the steps too

Answer 1

Step-by-step explanation:

Let f(x)=x+1xf(x)=x+1x. f(x)f(x) is defined for x≠0x≠0. It is not difficult to prove that if f(x)=f(y)f(x)=f(y) then either y=xy=x or y=1xy=1x. Now (a2+b2)/(c2+d2)=ab/cd(a2+b2)/(c2+d2)=ab/cd, if a,b,c,d≠0a,b,c,d≠0, can be rewritten as f(ab)=f(cd)f(ab)=f(cd). So it follows from the remark above that either ab=cdab=cd or ab=dcab=dc. If a=b=0a=b=0 and c,d≠0c,d≠0, the equality holds as well, but in this case it does not follow that a:b=c:da:b=c:d, however it does follow that