Question

If (a + b + c + d) (a - b - c + d) = (a + b - c - d) (a - b + c - d), prove that a: b = c: d​

Answer 1

Answer:

Given:

(a + b + c + d) (a - b - c + d) = (a + b - c - d) (a - b + c - d)

Simplifying left side:

(a + b + c + d) (a - b - c + d)=

a²-ab-ac+ad+ab-b²-bc+bd+ac-bc-c²+cd+ad-bd-cd+d²=

a²+2ad-b²-c²+d²

Simplifying right side:

(a + b - c - d) (a - b + c - d)=

a²-ab+ac-ad+ab-b²+bc-bd-ac+bc-c²+cd-ad+bd-cd+d²=

a²-2ad-b²+2bc-c²+d²

Now time to compare both sides:

a²+2ad-b²-c²+d²=a²-2ad-b²+2bc-c²+d²

2ad=-2ad+2bc

4ad=2bc

2ad=bc

2a:b=c:d, Proved

Answer 2

a^2 - b^2 - c^2 + d^2 = a^2 - b^2 + c^2 - d^2

a - b - c + d =a - b + c - d

2d=2c

d=c