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GATE 2024 EE – Question 5

General Aptitude · Quantitative Aptitude: Algebra, Geometry and Mensuration · 1 mark · Multiple choice

If, for non-zero real variables $x$, $y$, and real parameter $a>1$,

$$x:y=(a+1):(a-1),$$

then, the ratio $(x^2-y^2):(x^2+y^2)$ is

  1. $2a:(a^2+1)$
  2. $a:(a^2+1)$
  3. $2a:(a^2-1)$
  4. $a:(a^2-1)$

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Show answer and explanation

Correct answer: (A) $2a:(a^2+1)$

Explanation

Take $x=a+1$, $y=a-1$ (the ratio is unchanged by scaling). Then $x^2-y^2=4a$ and $x^2+y^2=2a^2+2$, so the ratio is $\dfrac{4a}{2(a^2+1)}=2a:(a^2+1)$.