Question

If a point $(x, y)$ moves on a curve not passing through the origin and satisfies the equation $\begin{vmatrix}a & b & a x+b y \\ b & c & b x+c y \\ a x+b y & b x+a y & 0\end{vmatrix}=0$, then $a, b, c$

• A. form an A.P.
• B. form a G.P.
• C. form a H.P.
• D. satisfy the relation $a ^2= b ^2+ c ^2$

Answer 1

Answer: (B)

$x\begin{vmatrix}a & b & a \\ b & c & b \\ a x+b y & b x+a y & 0\end{vmatrix}+y\begin{vmatrix}a & b & b \\ b & c & c \\ a x+b y & b x+a y & 0\end{vmatrix}$
$\Rightarrow \text { Apply } C _1 \rightarrow C _1- C _3 \text { and } C _2- C _3 $
$\Rightarrow\left( b ^2- ac \right)\left( ax ^2+ ay ^2+( a + b ) xy \right)=0 $
$\Rightarrow b ^2= ac ( GP ) $