Question

$x=$............... is identified as a GOLDEN RATIO

• A. $\frac{1+\sqrt{5}}{2}$
• B. $0$
• C. $\frac{1+\sqrt{2}}{2}$
• D. $1$

Answer 1

The correct option is A $\frac{1+\sqrt{5}}{2}$

Two quantities are said to be in golden ratio if their ratio is same as the ratio of their sum to the larger of the two quantities.

Let us take two numbers $a$ and $b$ where $a>b$

As per the definition, $\frac{a}{b}=\frac{a+b}{a}$

$\Rightarrow a\times a=b\times (a+b)$

$\Rightarrow a^2=ab+b^2$

$\therefore a^2-ab-b^2=0$

Assuming a quadratic equation in $a$, we get $a=\frac{b\pm \sqrt{b^2+4b^2}}{2}$

$\therefore a=\frac{b+\sqrt{5}b}{2}$

$\Rightarrow \frac{a}{b}=\frac{1+\sqrt{5}}{2}$

This is the ratio of the numbers we had taken and is defined as the golden ratio.