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Equation of Tangent at a Point (x,y) in Terms of f'(x)

Find the valu...

Question

Find the value of $k$ for which the expression $x^{2} + 2 x y + {k y}^{2} + 2 x + k = 0$ can be resolved into two linear factors.

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Solution

$x^{2} + 2 x y + {k y}^{2} + 2 x + k = 0$
(Pair of straight lines)
$Δ = 0$
$\Rightarrow 1 (k) (k) + 0$
$- 1 (0) - k (2) = 0$
$\Rightarrow k^{2} - 2 k = 0$
$k = 0, 2$

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