In the verge of coronavirus pandemic, we are providing FREE access to our entire Online Curriculum to ensure Learning Doesn't STOP!

Introduction to Pairs of Lines

Go back to  'Straight Lines'

Consider two lines

\[\begin{align} {L_1} \equiv y - {m_1}x - {c_1} = 0\\  {L_2} \equiv y - {m_2}x - {c_2} = 0  \end{align}\]

What do you think will \({L_1}{L_2} = 0\) represent? It is obvious that any point lying on \({L_1}\) and \({L_2}\) will satisfy \({L_1}{L_2} = 0,\) and thus \({L_1}{L_2} = 0\) represents the set of points constituting both the lines, i.e.,

\[\boxed{{L_1}{L_2} = 0{\text{ represents the pair of straight lines given by }}{L_1} = 0\,\,{\text{and }}{L_2} = 0}\]

For example, consider the equation \({y^2} - {x^2} = 0.\) What does this represent ? We have

\[\begin{align} \qquad & {y^2} - {x^2} = 0 \qquad \qquad \qquad \qquad...(1)\\  \Rightarrow \qquad & (y + x)(y - x) = 0\\   \Rightarrow \qquad  & (1)\;{\rm{represents}}\;{\text{the pair of straight lines}}\;x = y\;{\rm{and}}\;x + y = 0.  \end{align}\]

There is nothing special about considering a pair. We can similarly define the joint equation of \(n\) straight lines \({L_i} \equiv y - {m_i}x - {c_i} = 0\;(i = 1,\,2...,n)\) as

\[\begin{align} & {L_1}{L_2}...{L_n} = 0\\   \Rightarrow \qquad & (y - {m_1}x - {c_1})(y - {m_2}x - {c_2})...(y - {m_n}x - {c_n}) = 0 \qquad \qquad \qquad ...(2)  \end{align}\]

Any point lying on any of these n straight lines will satisfy (2), and thus (2) represents the set of all points constituting the n lines, i.e. (2) represents the joint equation of the n straight lines.

What is relevant to us at this stage is only a pair of straight lines and it is on a pair of lines that we now focus our attention.

Download SOLVED Practice Questions of Introduction to Pairs of Lines for FREE
Straight Lines
grade 11 | Answers Set 2
Straight Lines
grade 11 | Questions Set 1
Straight Lines
grade 11 | Answers Set 1
Straight Lines
grade 11 | Questions Set 2
Download SOLVED Practice Questions of Introduction to Pairs of Lines for FREE
Straight Lines
grade 11 | Answers Set 2
Straight Lines
grade 11 | Questions Set 1
Straight Lines
grade 11 | Answers Set 1
Straight Lines
grade 11 | Questions Set 2
Learn from the best math teachers and top your exams

  • Live one on one classroom and doubt clearing
  • Practice worksheets in and after class for conceptual clarity
  • Personalized curriculum to keep up with school