Question

The three sides of a right-angled triangle are in G.P. (geometrical progression). If the two acute angles be $\alpha$ and $\beta$, then $\tan \alpha$ and $\tan \beta$ are

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

Answer 1

The correct option is B $\sqrt{\frac{\sqrt{5}+1}{2}}$ and $\sqrt{\frac{\sqrt{5}-1}{2}}$

Let sides be $a,ar,a{r}^2(r>1)$
Using Pythagoras theorem
$a^2+a^2{r}^2=a^2{r}^4$
$\Rightarrow r^4-r^2-1=0$
$\Rightarrow r^2=\frac{1\pm \sqrt{5}}{2}\Rightarrow r=\sqrt{\frac{1+\sqrt{5}}{2}}$

Now,
$\tan \alpha ,\tan \beta$ equal to $r$ or $\frac{1}{r}$
$\Rightarrow \tan \alpha ,\tan \beta =\sqrt{\frac{\sqrt{5}+1}{2}}\text{and}\sqrt{\frac{\sqrt{5}-1}{2}}$