if `a^x=(a/k)^y=k^m`then prove that `1/x-1/y=1/m`

If 2x=y^(1/m)+y^(-1/m) , then prove that (x^2-1)y_(n+2)+(2n+1)xy_(n+1)+(n^2-m^2)y_n=0 .

If x/l+y/m=1 is any line passing through the intersection point of the lines x/a+y/b=1 and x/b+y/a=1 then prove that 1/l+1/m=1/a+1/b

If x/l+y/m=1 is any line passing through the intersection point of the lines x/a+y/b=1 and x/b+y/a=1 then prove that 1/l+1/m=1/a+1/b

If (x)/(l)+(y)/(m)=1 is any line passing through the intersection point of the lines (x)/(a)+(y)/(b)=1 and (x)/(b)+(y)/(a)=1 then prove that (1)/(l)+(1)/(m)=(1)/(a)+(1)/(b)

If y^(1//m)+y^(-1//m)=2x , then prove that (x^(2)-1)y_(2)+xy_(1)=m^(2)y^(2)

If y=e^(m sin^(-1)x) then prove that (1-x^2) y_2 - xy_1 = m^2 y

If y= (x + sqrt(x^(2) + 1))^(m) then prove that, (x^(2)+1) y_(2) + xy_(1)= m^(2)y

If sin^(-1)((x^(2)-y^(2))/(x^(2)+y^(2)))=k , k is a constant, then prove that (dy)/(dx)=(y)/(x) .

If sin^(-1) ((x^2-y^2)/(x^2+y^2))=k , k is a constant, then prove that (dy)/(dx)=y/x .

If y is the H.M. between x and z, prove that, (1)/(y-x) + (1)/(y-z) = (1)/(x) + (1)/(z) .