Question

The equation
$ax+a^{2}y+z=0$
$bx+b^{2}y+z=0$
$cx+c^{2}y+z=0$
have only one solution $(0,0,0)$
The coefficient of a,b,c are in G.P then the common ratio of G.P cannot be equal to:-

• A. $0,\pm 1$
• B. $2,\pm 3$
• C. $4,\pm 5$
• D. none of these

Answer 1

Answer: (A)

$0,\pm 1$

Given $a,b,c$ are in G.P.

let the common ration be $r$

$⟹b=ar,c=a{r}^2$

$ax+a^{2}y+z=0$

$bx+b^{2}y+z=0$

$cx+c^{2}y+z=0$

for these system of equations to have only one solution$(0,0,0)$

$\left|\begin{array}{ccc}a& a^2& 1\\ b& b^2& 1\\ c& c^2& 1\end{array}\right|\ne 0$

$(a-b)(b-c)(c-a)\ne 0$

$(a-ar)(ar-a{r}^2)(a{r}^2-a)\ne 0$

$a^{3}r(1-r{)}^2(r^2-1)\ne 0$

$r\ne 0,-1,1$

$r=0,\pm 1$