Question

if tan theta and cot theta are the roots of the equation
${x}^{2} + px + q = 0$
then the value of q is​

Answer 1

q is 1

as

q= product of zeros

= tan theta*cot theta

=tan theta*1/tan theta

1

Answer 2

Given:

$x^{2} +px+q=0$

Tanθ and Cotθ are roots of the equation.

To find:

Value of $q$.

Solution:

An equation $ax^{2} +bx+c=0$ has roots $\alpha$ and $\beta$ where sum and product of the roots are given by:

$\alpha +\beta =\frac{-b}{a}$

$\alpha \beta =\frac{c}{a}$

Here, we have $a=1,b=p,c=q$

Let $\alpha =tan\theta$

$\beta =Cot\theta$

Sum of roots, $\alpha +\beta =\frac{-b}{a}$

$Tan\theta+Cot\theta=\frac{-p}{1}$

∴ $p=-(Tan\theta+Cot\theta)$

Product of roots, $\alpha \beta =\frac{c}{a}$

$Tan\theta Cot\theta=\frac{q}{1}$

∴ $q=Tan\theta Cot\theta$

But, $Cot\theta=\frac{1}{Tan\theta}$

Hence, $q=Tan\theta*\frac{1}{Tan\theta}$

∴ $q=1$

Value of q is 1.