First slide

Pair of straight lines

Question

If the equation $k^{2} x^{2} + 2 xy + 9 y^{2} = 0$ represents a pair of distinct lines, then k lies in the internal

Easy

A

$[\frac{- 1}{3}, \frac{1}{3})$
B

$(\frac{- 1}{3}, \frac{1}{3}]$
C

$(\frac{- 1}{3}, \frac{1}{3})$
D

$[\frac{- 1}{3}, \frac{1}{3}]$

Solution

Detailed solution:
The condition that the equation ${ax}^{2} + 2 hxy + {by}^{2} = 0$ to represents a pair of lines is $h^{2} > ab$
Here $a = k^{2}, h = 1, b = 9$
Hence,
$\begin{matrix} 1^{2} > k^{2} (9) \\ k^{2} < \frac{1}{9} \\ |k| < \frac{1}{3} \end{matrix}$
Therefore, the interval where the value of lies is $k \in (- \frac{1}{3}, \frac{1}{3})$

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