If the lines passing through the points $(2, 3), (K^{2}, 6)$ and $(6, 4), (K, 6)$ are parallel, then the total number of possible real values of $K$ is

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Solution

$m_{1} =$ Slope of line passing through $(2, 3), (K^{2}, 6)$
$= \frac{3}{K^{2} - 2}$
$m_{2} =$ Slope of line passing through $(6, 4), (K, 6)$
$= \frac{2}{K - 6}$
As lines are parallel. So
$m_{1} = m_{2} \Rightarrow \frac{3}{K^{2} - 2} = \frac{2}{K - 6}$
$\Rightarrow 3 K - 18 = 2 K^{2} - 4$
$\Rightarrow 2 K^{2} - 3 K + 14 = 0$
As $D < 0$ , so no real value of $K$ is possible.

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