Question

If a and b are the roots of $x^2-3x+p=0$ and c, d are the roots $x^2-12x+q=0$, Where a, b, c, d form a G.P. Prove that (q + p) : (q - p) = 17 : 15.

Answer 1

Given,

a, b are roots of the equation $x^2-3x+p=0$ $\Rightarrow a+b=3,$ ab = p

and c, d are the roots of the equation $x^2+12x+q=0$

$\Rightarrow$ Let $b=ar,c=a{r}^{2}andd=a{r}^3,$ then a + b = 3 and c + d = 12

a(1 + r) = 3 and $a{r}^2(1+r)=12$

$\Rightarrow \frac{a{r}^2(1+r)}{a(1+r)}=\frac{12}{3}$

$\Rightarrow$ r = 2

and a(r + 1) = 3

$\Rightarrow$ a = 1

$\Rightarrow$ p = ab = a $\times$ ar

$\Rightarrow$ p = 2

$\Rightarrow$ q = cd = $a{r}^2\times a{r}^3$

$\Rightarrow$ q = $2^5$

$q=32$

$\therefore \frac{q+p}{q-p}=\frac{32+2}{32-2}$

$=\frac{34}{30}$

$\therefore$ (q + p) : (q - p) = 17 : 15