Question

If the ratio of the roots of the equation $x^2+bx+c=0$ is the same as that of $x^2+qx+r=0$, then

• A. $r^{2}b=q{c}^2$
• B. $r^{2}c=q{b}^2$
• C. $c^{2}r=q^{2}b$
• D. $b^{2}r=q^{2}c$

Answer 1

The correct option is D $b^{2}r=q^{2}c$

$x^2+bx+c=0$ ...(1)
$x^2+qx+r=0$ ...(2)
Let $\alpha$ and $\beta$ be the roots of the equation (1)
Let $\gamma$ and $\delta$ be the roots of equation (2)
It is given that $\frac{\alpha }{\beta }=\frac{\gamma }{\delta }$ ...(3)
$\alpha =k\gamma$ and $\beta =k\delta$ ($k$ is a constant)

Now, $\alpha \beta =c$
$\therefore c=k^2\gamma \delta =k^{2}r$ ...(4)

Also, $\alpha +\beta =-b$
$\therefore -b=k(\gamma +\delta )=-kq$ ...(5)
Therefore from (4) and (5) we get
$\frac{b^2}{q^2}=\frac{c}{r}=k^2$
Therefore From the above equation
$b^{2}r=c{q}^2$