Matrices And Determinants Ques 5
- Let $a, b, c$ be real numbers with $a^2+b^2+c^2=1$. Show that the equation
$\left|\begin{array}{ccc}a x-b y-c & b x+a y & c x+a \\ b x+a y & -a x+b y-c & c y+b \\ c x+a & c y+b & -a x-b y+c\end{array}\right|=0$ represents a straight line.
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Solution:
$ \text { Given, }\left|\begin{array}{ccc} a x-b y-c & b x+a y & c x+a \\ b x+a y & -a x+b y-c & c y+b \\ c x+a & c y+b & -a x-b y+c \end{array}\right|=0 $
$ \Rightarrow \frac{1}{a}\left|\begin{array}{ccc} a^2 x-a b y-a c & b x+a y & c x+a \\ a b x+a^2 y & -a x+b y-c & c y+b \\ a c x+a^2 & c y+b & -a x-b y+c \end{array}\right|=0 $
Applying $C_1 \rightarrow C_1+b C_2+c C_3$
$ \Rightarrow \frac{1}{a}\left|\begin{array}{ccc} \left(a^2+b^2+c^2\right) x & a y+b x & c x+a \\ \left(a^2+b^2+c^2\right) y & b y-c-a x & b+c y \\ a^2+b^2+c^2 & b+c y & c-a x-b y \end{array}\right|=0 $
$\Rightarrow \quad \frac{1}{a}\left|\begin{array}{ccc}x & a y+b x & c x+a \\ y & b y-c-a x & b+c y \\ 1 & b+c y & c-a x-b y\end{array}\right|=0$
$ \left[\because a^2+b^2+c^2=1\right] $
Applying $C_2 \rightarrow C_2-b C_1$ and $C_3 \rightarrow C_3-c C_1$
$\Rightarrow \quad \frac{1}{a}\left|\begin{array}{ccc}x & a y & a \\ y & -c-a x & b \\ 1 & c y & -a x-b y\end{array}\right|=0$
$\Rightarrow \quad \frac{1}{a x}\left|\begin{array}{ccc}x^2 & a x y & a x \\ y & -c-a x & b \\ 1 & c y & -a x-b y\end{array}\right|=0$
Applying $R_1 \rightarrow R_1+y R_2+R_3$
$\Rightarrow \quad \frac{1}{a x}\left|\begin{array}{ccc}x^2+y^2+1 & 0 & 0 \\ y & -c-a x & b \\ 1 & c y & -a x-b y\end{array}\right|=0$
$\Rightarrow \quad \frac{1}{a x}\left[\left(x^2+y^2+1\right)\{(-c-a x)(-a x-b y)-b(c y)\}\right]=0$
$\Rightarrow \quad \frac{1}{a x}\left[\left(x^2+y^2+1\right)\left(a c x+b c y+a^2 x^2+a b x y-b c y\right)\right]=0$
$\Rightarrow \quad \frac{1}{a x}\left[\left(x^2+y^2+1\right)\left(a c x+a^2 x^2+a b x y\right)\right]=0$
$\Rightarrow \quad \frac{1}{a x}\left[a x\left(x^2+y^2+1\right)(c+a x+b y)\right]=0$
$\Rightarrow \quad \left(x^2+y^2+1\right)(a x+b y+c)=0$
$\Rightarrow \quad a x+b y+c=0$
which represents a straight line.
BETA
