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If $a x^{2}-y^{2}+4 x-y=0$ represents a pair of lines, then $a$ is equal to
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Verified Answer
The correct answer is:
$16$
$a x^{2}-y^{2}+4 x-y=0$ represents a pair of straight line, if
$$
\begin{aligned}
&\left|\begin{array}{ccc}
a & 0 & 2 \\
0 & -1 & -1 / 2 \\
2 & -1 / 2 & 0
\end{array}\right|=0 \\
\Rightarrow \quad a\left(0-\frac{1}{4}\right)+2(0+\overline{2}) &=\overline{0} \\
\Rightarrow \quad-\frac{a}{4}+4 &=0 \\
a &=16
\end{aligned}
$$
$$
\begin{aligned}
&\left|\begin{array}{ccc}
a & 0 & 2 \\
0 & -1 & -1 / 2 \\
2 & -1 / 2 & 0
\end{array}\right|=0 \\
\Rightarrow \quad a\left(0-\frac{1}{4}\right)+2(0+\overline{2}) &=\overline{0} \\
\Rightarrow \quad-\frac{a}{4}+4 &=0 \\
a &=16
\end{aligned}
$$
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