For ℓ ∈ R , the equation (2 ℓ-3) x^2+2 ℓ xy-y^2=0 represents a pair of distinct lines

Correct answer: (D) for all values of ℓ ∈ R-[-3,1]. a x^2+2 ~h x y+b y^2=0 represents distinct lines if h^2-a b>0 ∴ l^2+(2 l-3)>0 l^2+2 l-3>0 (l+3)(l-1)>0 ∴ l ∈(-∞,-3) ∪(1, ∞) or l ∈ R-[-3,1]

Q87 / 160 · TS EAMCET 2023 (14 May Shift 1) · Mathematics

For $\ell \in \mathbb{R}$, the equation $(2 \ell-3) \mathrm{x}^2+2 \ell \mathrm{xy}-\mathrm{y}^2=0$ represents a pair of distinct lines

A
only whən $\ell=0$
B
for all values of $\ell \in(-3,1)$
C
for all values of $\ell \in \mathbb{R}-(0,1)$
Correct answer
D
for all values of $\ell \in \mathbb{R}-[-3,1]$

Solution

$a x^2+2 \mathrm{~h} x y+\mathrm{b} y^2=0$ represents distinct lines if $$ \begin{aligned} & h^2-a b>0 \therefore l^2+(2 l-3)>0 \\ & l^2+2 l-3>0 \\ & (l+3)(l-1)>0 \\ & \therefore l \in(-\infty,-3) \cup(1, \infty) \text { or } l \in \mathrm{R}-[-3,1] \end{aligned} $$

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

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