What should be the value of $c$ , so that the expression $2 x^{2} - 3 x y + c y^{2} + 5 x + 2$ can be resolved into product of two linear factors ?

2

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3

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Solution

The correct option is C $- 2$
$Δ = ∣ ∣ ∣ ∣ \begin{matrix} a & h & g h & b & f g & f & c \end{matrix} ∣ ∣ ∣ ∣$
A general equation, $a x^{2} + 2 h x y + b y^{2} + 2 g x + 2 f y + c = 0$
represents a pair of lines only if $△ = 0$
where $△$ is defined above.
comparing with values in question we have,
$a = 2, h = - 1.5, g = 2.5, b = c, f = 0, c = 2$
Solving the determinant after putting values gives $c = - 2$

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