Q61 / 160 · TS EAMCET 2023 (12 May Shift 1) · Mathematics

The range of ' $a$ ' so that $a^2 x^2+2 x y+4 y^2=0$ represents two distinct lines is

A
$a>\frac{1}{2}$ or $a < \frac{-1}{2}$
B
$\frac{-1}{2} \leq a \leq \frac{1}{2}$
Correct answer
C
$\frac{-1}{2} < a < \frac{1}{2}$
D
$a \geq \frac{1}{2}$ or $\leq \frac{-1}{2}$

Solution

$a^2 x^2+2 x y+4 y^2=0$ $\Rightarrow a x^2+2 h x y+b y^2=0$ represents pair of straight lines if $h^2-a b>0$ $\begin{aligned} & 1-4 a^2>0 \\ \Rightarrow & 4 a^2-1 < 0 \\ \Rightarrow & (2 a+1)(2 a-1) < 0 \\ \therefore & \frac{-1}{2} < a < \frac{1}{2} . \end{aligned}$

Hence, the correct answer is option: 'C'.

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