If the lines represented by $x^{2} + k x y + 4 y^{2}$ are coincident, then find the value of $k$ .

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Solution

The given equation is
$x^{2} + k x y + 4 y^{2}$ ......(i)
Comparing with $a x^{2} + 2 h x y + b y^{2}$ , we get
$a = 1, b = 4, 2 h = k$ $\Rightarrow$ $h = \frac{k}{2}$
If the lines represented by (i) are coincident, then
$h^{2} = a b$
$\Rightarrow {(\frac{k}{2})}^{2} = (1) (4)$
$\Rightarrow$ $k^{2} = 16$
$\Rightarrow$ $k = \pm 4$

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