Question

If `(x + y)/(ax + by) = (y + z)/(ay + bz) = (z + x)/(az + bx)`, prove that each of these ratio is equal to `(2)/(a + b)` unless x + y + z = 0.

Answer 1

`(x + y)/(ax + by) = (y + z)/(ay + bz) = (z + x)/(az + bx)`

= `((x + y) + (y + z) + (z + x))/((ax + by) + (ay + bz) + (az + bx))`

Numerator:

= (x + y) + (y + z) + (z + x)

= 2x + 2y + 2z = 2(x + y + z)

Denominator:

= (ax + by) + (ay + bz) + (az + bx)

= ax + ay + az + bx + by + bz

= a(x + y + z) + b(x + y + z)

= (a + b) (x + y +z)

Substitute these back into the expression:

= `(2(x + y + z))/((a + b)(x + y + z)`

= `2/(a + b)`

if x + y + z ≠ 0.

Hence proved.