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Question

Show that the equation $$x^{2}+2xy+2y^{2}+2x+2y+1=0$$ does not represent a pair of lines.


Solution

Comparing the equation
$$x^{2}+2xy+2y^{2}+2x+2y+1=0$$ with
$$ax^{2}+2hxy+by^{2}+2gx+2fy+c=0$$, we get,
$$a=1, h=1, b=2, g=1, f=1, c=1$$.
The given equation represents a pair of lines, if
$$D=\begin{vmatrix} a & h & g \\ h & b & f \\ g & f & c \end{vmatrix}=0$$ and $$h^{2}-ab \ge 0$$
Now, $$D=\begin{vmatrix} a & h & g \\ h & b & f \\ g & f & c \end{vmatrix}=\begin{vmatrix} 1 & 1 & 1 \\ 1 & 2 & 1 \\ 1 & 1 & 1 \end{vmatrix}$$
$$=1(2-1)-1(1-1)+1(1-2)$$
$$=1-0-1=0$$
and $$h^{2}-ab=(1)^{2}-1(2)=-1 < 0$$
$$\therefore$$ given equation does not represent a pair of lines.

Maths

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