Question

If $3\text{and}4$ lies between the roots of the equation $x^2+2kx+9=0$ then $k$ lies in the interval

• A. $(-\frac{25}{8},\infty )$
• B. $(-\infty ,-\frac{25}{8})\cup (3,\infty )$
• C. $(-\infty ,-3)$
• D. $(-\infty ,-\frac{25}{8})$

Answer 1

The correct option is D $(-\infty ,-\frac{25}{8})$

Let $f\left(x\right)=x^2+2kx+9$, whose roots are $\alpha ,\beta$
Given: $\alpha <3<4<\beta$

Conditions:

$\left(i\right)f\left(3\right)<0\Rightarrow 6k+18<0\Rightarrow k<-3\dots \left(1\right)$

$\left(ii\right)f\left(4\right)<0\Rightarrow 8k+25<0\Rightarrow k<-\frac{25}{8}\dots \left(2\right)$

From $\left(1\right),\left(2\right)$, we get
$k\in (-\infty ,-\frac{25}{8})$