Question

Prove that `(x/a)^n+(y/b)^n=2` touches the straight line `x/a+y/b=2` for all ` in N` , at the point `(a , b)` .

Answer 1

`n(x/a)^(n-1)*1/a+n(y/b)^(n-1)*1/b*dy/dx=0`
`dy/dx=(-n(x/a)^(n-1)*(1/a))/(n(y/b)^(n-1)*(1/b)`
`=(-b(x/a)^(n-1))/(a(y/b)^(n-1))`
`-b/a((x/a)^(n-1)/(y/b)^(n-1))=-b/a`
`y-b=-b/a(x-a)`
`y/b-1=-x/a+1`
`x/a+y/b=2`.