GATE Electrical 2024 | GA | Question: 5

Question

​​If, for non-zero real variables $x, y$, and real parameter $a>1$,
\[
x: y=(a+1):(a-1) \text {, }
\]
then, the ratio $\left(x^{2}-y^{2}\right):\left(x^{2}+y^{2}\right)$ is

• A. $2 a:\left(a^{2}+1\right)$
• B. $a:\left(a^{2}+1\right)$
• C. $2 a:\left(a^{2}-1\right)$
• D. $a:\left(a^{2}-1\right)$

Comments

2a:a^2+1

---

first do the sqaure on both sides of given eqn,

then

apply componendo and dividendo.....

x/y = a/b

x+y/x-y = a+b/a-b

Answer 1

Let, $K$ be a real number such that

\[
\frac{x^2}{y^2}=\frac{(a+1)^2}{(a-1)^2}=K
\]
$\implies$
\[
{x^2}=Ky^2
\]
$\implies$
\[
\frac{x^2-y^2}{x^2+y^2}=\frac{y^2(K-1)}{y^2(K+1)}=\frac{\frac{(a+1)^2}{(a-1)^2} -1}{\frac{(a+1)^2}{(a-1)^2} +1}
\]
\[
\frac{x^2-y^2}{x^2+y^2}=\frac{(a^2+2a+1)-(a^2-2a+1)}{(a^2+2a+1)+(a^2-2a+1)}
\]
\[
\frac{x^2-y^2}{x^2+y^2}=\frac{4a}{2(a^2+1)}=\frac{2a}{a^2+1}
\]

Hence, Option A is the correct answer.

Comments

Substitution of K can be added as a step for more clarification

---

Apply Componendo and Dividendo Rule.

https://www.geeksforgeeks.org/componendo-dividendo-rule/

Answer 2

Correct ans is option(A)

Answer 3

Option:A is the answer