Question

Show that (A•B)2 + (Ax B)2 = A²B^2.​

Answer 1

Let x/a + y/b = 2 ··· Eq.1, and ax - by = a² - b² ··· Eq.2

Multiply Eq.1 by a²b, and Eq.2 by b. Eq.1 and Eq.2 can be rewritten as follows, respectively:

abx + a²y = 2a²b ··· Eq.3,

abx - b²y = a²b - b³ ··· Eq.4 Subtract Eq.4 from Eq.3.

(a² + b²)y = 2a²b - (a²b - b³) ⇒ (a² + b²)y = a²b + b³ ⇒ (a² + b²)y = (a² + b²)b We have: y = b

Plug y = b into Eq.1 x/a + b/b = 2 ⇒ x/a + 1 = 2 ⇒ x/a = 1 We have: x = a

CK: Plug x = a and y = b into LHS of Eq.2. LHS = ax - by = a² - b², while RHS = a² - b²,

so LHS = RHS CKD.

The answer is: x = a, and y = b **