Question

𝑥 : 𝑦 = 3 : 2 and \[(𝑥^2 + 𝑦^2) : (𝑥^2 − 𝑦^2)\].

Assertion (A): The value of ratio \[(𝑥^2 + 𝑦^2) : (𝑥^2 − 𝑦^2)\] = 13 : 5.

Reason (R): 𝑥 : 𝑦 = 3 : 2 \[\Rightarrow \frac{x^{2} + y^{2}}{x^{2} - y^{2}} = \frac{(3k)^{2} + (2k)^{2}}{(3k)^{2} - (2k)^{2}} = \frac{13}{5};\ k \neq 0.\]

Options

• A is true, R is false.

• A is false, R is true.

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

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

Answer 1

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

Explanation:

Given,

x : y = 3 : 2

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

The value of

⇒ `(x^2−y^2)/(x^2+y^2)​`

= `((3k)^2−(2k)^2)/((3k)^2+(2k)^2​)`

= `(9k^2−4k^2)/(9k^2+4k^2)​`

= `(13k^2)/(5k^2)`

= `13/5`

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

According to Reason; x : y = 3 : 2

\[\Rightarrow \frac{x^{2} + y^{2}}{x^{2} - y^{2}} = \frac{(3k)^{2} + (2k)^{2}}{(3k)^{2} - (2k)^{2}} = \frac{13}{5};\ k \neq 0.\] which is true.