TS EAMCET · Maths · Pair of Lines
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 \(\ell \in \mathbb{R}-(0,1)\)
- D for all values of \(\ell \in \mathbb{R}-[-3,1]\)
Answer & Solution
Correct Answer
(D) for all values of \(\ell \in \mathbb{R}-[-3,1]\)
Step-by-step Solution
Detailed explanation
\(a x^2+2 \mathrm{~h} x y+\mathrm{b} y^2=0\) represents distinct lines if \[ \begin{aligned} & h^2-a b>0 \therefore l^2+(2 l-3)>0 \\ & l^2+2 l-3>0 \\ & (l+3)(l-1)>0 \\ & \therefore l \in(-\infty,-3) \cup(1, \infty) \text { or } l \in \mathrm{R}-[-3,1] \end{aligned} \]
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