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JEE Advanced
Exam
JEE Advanced
Subject
Mathematics
Chapter
Hyperbola

JEE Advanced · Mathematics · Hyperbola

If the following represent the two lines with real point of...

If the following represent the two lines with real point of intersection then α1x+α2y=k1\alpha_1 x+\alpha_2 y=k_1, β1x+β2y=k2\beta_1 x+\beta_2 y=k_2. Also, (α1,α2),(β1,β2)\left(\alpha_1, \alpha_2\right), \left(\beta_1, \beta_2\right) lie on x2+y2=1x^2+y^2=1 then

Options

  1. Aα1β1+α2β2∈[0,2]\alpha_1 \beta_1+\alpha_2 \beta_2 \in[0,2]
  2. Blines can represent asymptotes of rectangular hyperbola
  3. Cα1β1+α2β2∈[−2,0]\alpha_1 \beta_1+\alpha_2 \beta_2 \in[-2,0]
  4. Dnone of these

Correct answer

B. lines can represent asymptotes of rectangular hyperbola

Step-by-step solution

\left|\alpha_1 \beta_1+\alpha_2 \beta_2\right| \leq \sqrt{\alpha_1^2+\alpha_1^2} \sqrt{\beta_1^2+\beta_2^2}\Rightarrow\left|\alpha_1 \beta_1+\alpha_2 \beta_2\right| \leq 1
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