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BITSAT 2020
Exam
BITSAT
Year
2020
Subject
Mathematics
Chapter
Pair of Lines

BITSAT 2020 · Mathematics · Pair of Lines

Statement- 1 : If the general equation x^ 2 +y^ 2 +2 x y+2 g x+2 f...

Statement- 1:Ifthegeneralequation: If the general equationx^{2}+y^{2}+2 x y+2 g x+2 f y+4=0representsapairofreallinesthenrepresents a pair of real lines then|g\mathrm{g}| \geq 2.Statement−2. Statement- 2: The equation ax2+2hxy+by2+2gx+2fy+c=0a x^{2}+2 h x y+b y^{2}+2 g x+2 f y+c=0 represents pair of real lines if abc+2fgh−af2−bg2−ch2=0a b c+2 f g h-a f^{2}-b g^{2}-c h^{2}=0

Options

  1. AStatement -1 is false, Statement- 2 is true
  2. BStatement -1 is true, Statement- 2 is true; Statement -2 is a correct explanation for Statement-1
  3. CStatement -1 is true, Statement- 2 is true; Statement -2 is not a correct explanation for Statement-1
  4. DStatement -1 is true, Statement- 2 is false

Correct answer

D. Statement -1 is true, Statement- 2 is false

Step-by-step solution

The equation represents a pair of lines if 1⋅1⋅4+2⋅f⋅g⋅1−1⋅\mathrmf2−1⋅\mathrmg2−4⋅12=0⇒(f−g)2=0⇒f=g1 \cdot 1 \cdot 4+2 \cdot \mathrm{f} \cdot \mathrm{g} \cdot 1-1 \cdot \mathrmf^{2}-1 \cdot \mathrmg^{2}-4 \cdot 1^{2}=0\Rightarrow(f-g)^{2}=0 \Rightarrow f=g The equation becomes (x+y)2+2g(x+y)+4=0(x+y)^{2}+2 g(x+y)+4=0 Which represents pair of parallel lines, which are real provided (2g)2−4⋅4⋅1≥0⇒∣g∣≥2(2 \mathrm{ g})^{2}-4 \cdot 4 \cdot 1 \geq 0 \Rightarrow|\mathrm{g}| \geq 2
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