If (x - 2y) (x + 2y) = 5 and (2x - y) (2x + y) = 35, then (x² - y²)/(x² + y²) =

Question: If (x - 2y) (x + 2y) = 5 and (2x - y) (2x + y) = 35, then (x² - y²)/(x² + y²) =

Solution:
Given,
(x - 2y) (x + 2y) = 5
⇒ x² - (2y)² = 5
⇒ x² - 4y² = 5..............(1)
Again,
(2x - y) (2x + y) = 35
⇒ (2x)² - y² = 35
⇒ 4x² - y² = 35..........(2)

Let, x² = a and y² = b
The equations becomes,
a - 4b = 5.........(3)
4a - b = 35........(4)
Now multiplying equation number (3) by 4,
4a - 16b = 20.........(5)

Now, subtract equation number (5) from equation number(4)
(4a - b) - (4a - 16b) = 35 - 20
⇒ 4a - b - 4a + 16b = 15
⇒ 15b = 15
∴ b = 1 [y² = 1]

Applying the value of b into the equation number (3),
a - (4 × 1) = 5
⇒ a - 4 = 5
⇒ a = 5 + 4
∴ a = 9 [x² = 9]

Now,
(x² - y²)/(x² + y²)
= (9 - 1)/(9 + 1)
= 8/10
= 4/5