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Q.
If the roots of the quadratic equation $k x^2+27 x$ $+20=0$ where $k \neq 0$ are in the ratio $4: 5$, then find the possible value of ' $K$ '.
Quadratic Equations
Solution:
Roots are in the ratio $4: 5$
Let $\alpha=4 P$
$\beta=5 P$
Sum of roots
$\alpha+\beta =\frac{-b}{a} $
$4 P+5 P =\frac{-27}{k}$
$9 P =\frac{-27}{k} $
$P =\frac{-3}{k}$
Product of roots
$ \alpha \cdot \beta=\frac{c}{a} $
$ \Rightarrow(4 P)(5 P)=\frac{20}{k} \Rightarrow 20 P^2=\frac{20}{k}$
$\Rightarrow\left(\frac{-3}{k}\right)^2=\frac{1}{k} \Rightarrow \frac{9}{k^2}=\frac{1}{k} \Rightarrow k=9$