x : y = 3 : 2 and (x² + y²) : (x² - y²)

Assertion (A) : The value of (x² + y²) : (x² - y²) = 13 : 5

Reason (R) : x : y = 3 : 2

⇒ $\dfrac{x^2 + y^2}{x^2 - y^2} = \dfrac{(3k)^2 + (2k)^2}{(3k)^2 - (2k)^2} = \dfrac{13}{5}$ ; k ≠ 0

1. A is true, R is false.

2. A is false, R is true.

3. Both A and R are true and R is the correct reason for R.

4. Both A and R are true and R is the incorrect reason for R.

Both A and R are true and R is the correct reason for R.

Reason

Given,

x : y = 3 : 2

Let the value of x be 3k and y be 2k.

The value of

$\Rightarrow\dfrac{x^2 + y^2}{x^2 - y^2}\\[1em] = \dfrac{(3k)^2 + (2k)^2}{(3k)^2 - (2k)^2}\\[1em] = \dfrac{9k^2 + 4k^2}{9k^2 - 4k^2}\\[1em] = \dfrac{13k^2}{5k^2}\\[1em] = \dfrac{13}{5}$

According to Assertion; the value of (x² + y²) : (x² - y²) = 13 : 5, which is true.

According to Reason; x : y = 3 : 2

⇒ $\dfrac{x^2 + y^2}{x^2 - y^2} = \dfrac{(3k)^2 + (2k)^2}{(3k)^2 - (2k)^2} = \dfrac{13}{5}$ ; k ≠ 0, which is true.

Hence, option 3 is the correct option.