The harmonic mean of two numbers is $-\frac{8}{5}$ and their geometric mean is 2 . The quadratic equation…

The harmonic mean of two numbers is $-\frac{8}{5}$ and their geometric mean is 2 . The quadratic equation whose roots are twice those numbers is
  1. $x^2+5 x+4=0$
  2. $x^2+10 x+16=0$
  3. $x^2-10 x+16=0$
  4. $x^2-5 x+4=0$

Solution

Let two numbers be $a$ and $b$ G.M. $=\sqrt{a b}=2$
Now, $\quad$ H.M. $=\frac{2 a b}{a+b}=-\frac{8}{5}$ $ \frac{2 \times 4}{a+b}=-\frac{8}{5} \quad \text { (from Eq. (i)) } $ $ \therefore \quad a+b=-5 $ Here, roots are twice of these numbers. So, $ \begin{aligned} & (2 a)(2 b)=4 a b=4 \times 4=16 \\ & 2 a+2 b=2(a+b)=2 \times-5=-10 \end{aligned} $ $\therefore \quad$ Quadratic Equation $x^2+10 x+16=0$

Asked in: AP EAMCET 2017 (26 Apr Shift 1)

Practice more Quadratic Equation questions on Aicharya