Equations of straight lines and circles - (12.1-12.4)

Understand the equation of a straight line

  • Theory
    1) Given the slope of a line L is m and it passes through a point \(\left( {{{\bf{x}}_1},\;{{\bf{y}}_1}} \right)\), where \({{\bf{x}}_1}\) and \({{\bf{y}}_1}\) are constants.
    The equation of L is given by

    \({\bf{L}}:\;\frac{{{\bf{y}} - {{\bf{y}}_1}}}{{{\bf{x}} - {{\bf{x}}_1}}} = m\)

  • Examples
    Given the slope of a line is -1 and it passes through ( 7 , -3 ),
    find the equation of the line.
  • Solutions

    The equation of the required line is,

    \(\frac{{y - \left( { - 3} \right)}}{{x - 7}} = - 1\)
    \(y + 3 = - x + 7\)
    \(x + y - 4 = 0\)

  • Theory
    2) Given a line L passes through points \(\left( {{{\bf{x}}_1},\;{{\bf{y}}_1}} \right)\) and \(\left( {{{\bf{x}}_2},\;{{\bf{y}}_2}} \right)\), where \({{\bf{x}}_1}\), \({{\bf{x}}_2}\), \({{\bf{y}}_1}\) and \({{\bf{y}}_2}\) are constants.
    The equation of L is given by

    \({\bf{L}}:\;\frac{{{\bf{y}} - {{\bf{y}}_1}}}{{{\bf{x}} - {{\bf{x}}_1}}} = \frac{{{{\bf{y}}_2} - {{\bf{y}}_1}}}{{{{\bf{x}}_2} - {{\bf{x}}_1}}}\)

  • Examples
    Find the equation of the straight line passing through A(10,-2) and B(4, 1).
  • Solutions
    By the two-point form,
    the equation of the straight line is
    \(\begin{array}{1}y - 1 = \frac{{ - {\rm{ }}2 - 1}}{{10 - 4}}(x - 4)\\y - 1 = \frac{{ - {\rm{ }}1}}{2}{(x - 4)^{}}\\2y - 2 = - x + 4\\x+ 2y - 6 = 0\end{array}\)
  • Theory
    3) Given L is a straight line and its equation is given by

    \({\bf{L}}:\;{\bf{y}} = {\bf{mx}} + {\bf{c}}\)

    then, m is the slope of the line and c is the y intercept of the lines
  • Examples
    Given \(L:x - 2y - 1 = 0\), find the slope, y – intercept and x – intercept of L.

  • Solutions
    \(x - 2y - 1 = 0\)
    \(y = \frac{1}{2}x - \frac{1}{2}\)
    \(\therefore m = \frac{1}{2}\)
    \(c = - \frac{1}{2}\)
    Put \(y = 0,\;x = 1\)

    \(\therefore \) the slope, y – intercept and x – intercept of L are
    \(\frac{1}{2}\) ,\( - \frac{1}{2}\) and 1 respectively