Question

Suppose $a^2=5a-8$ and $b^2=5b-8$, then equation whose roots are $\frac{a}{b}$ and $\frac{b}{a}$ is

• A. $8x^2+9x+8=0$
• B. $9x^2-8x+9=0$
• C. $8x^2-9x+8=0$
• D. $6x^2-5x-6=0$

Answer 1

The correct option is C

$8x^2-9x+8=0$

Given: $a^2=5a-8$ and $b^2=5b-8$

$\Rightarrow a^2-5a+8=0$ and $b^2-5b+8=0$

$\Rightarrow a,b$ are roots of $x^2-5x+8=0$

$\therefore$ $a+b=5,ab=8$

$\Rightarrow \frac{a}{b}+\frac{b}{a}=\frac{a^2+b^2}{ab}=\frac{(a+b{)}^2-2ab}{ab}$

$=\frac{25}{8}-2=\frac{9}{8}$

Thus, the required equation is

$x^2-\left(\frac{9}{8}\right)x+1=0$

$\Rightarrow 8x^2-9x+8=0$