Question

If $\tan a,\tan b$ are the roots of $x^2-px+q=0$ and $\cot a,\cot b$ are the roots of $x^2-rx+s=0$, then the value of $rs$ is

• A. $\frac{p}{q^2}$
• B. $\frac{p}{q}$
• C. $p+q$
• D. $\frac{q^2}{p}$

Answer 1

The correct option is A $\frac{p}{q^2}$

$x^2-px+q=0$
$\tan a+\tan b=p,\tan a\tan b=q$
$x^2-rx+s=0$
$\cot a+\cot b=r,\cot a\cot b=s$

Now,
$rs=(\cot a+\cot b)\cot a\cot b$
$=(\frac{1}{\tan a}+\frac{1}{\tan b})\frac{1}{\tan a\tan b}$
$=\frac{\tan a+\tan b}{(\tan a\tan b{)}^2}=\frac{p}{q^2}$