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# For the equation $ax^2+2hxy+by^2+2gx+2fy+c=0$ to be parabola what should be the conditions?

$\begin{array}{1 1}(a)\;\Delta\neq 0\;and\;ab-h^2=0\\(b)\;\Delta\neq 2\;and\;ab-2h^2=0\\(c)\;\Delta\neq 4\;and\;ab-h=0\\(d)\;\Delta\neq 0\;and\;ab^2-h=0\end{array}$

$\Delta \neq 0$ where $\Delta=abc+2fgh-af^2-bg^2-ch^2$
$ab-h^2=0$
When these conditions are satisfied then this equation will represent a parabola.
Hence (a) is the correct answer.