Equation of a Line; Section Formula

A line $AB$ meets the X-axis at $A$ and the Y-axis at $B$. $P \left(4, -1\right)$ divides $AB$ in the ratio $1 : 2$.

  1. Find the coordinates of $A$ and $B$.
  2. Find the equation of the line through $P$ and perpendicular to $AB$.


  1. Given: $\;$ Line $AB$ meets the X-axis at $A$ and the Y-axis at $B$.

    $\therefore \;$ Let the coordinates of point $A$ be $\left(a, 0\right)$

    and the coordinates of point $B$ be $\left(0, b\right)$.

    Given: $\;$ $P \left(4, -1\right)$ divides $AB$ in the ratio $1 : 2$

    $\therefore \;$ We have by section formula,

    $4 = \dfrac{1 \times 0 + 2 \times a}{1 + 2}$

    i.e. $\;$ $12 = 2a$ $\implies$ $a = 6$

    and $\;$ $-1 = \dfrac{1 \times b + 2 \times 0}{1 + 2}$ $\implies$ $b = -3$

    $\therefore \;$ $A = \left(6, 0\right)$; $\;$ $B = \left(0, -3\right)$


  2. Slope of $AB = m_1 = \dfrac{-3 - 0}{0 - 6} = \dfrac{1}{2}$

    Slope of line perpendicular to $AB = m_2 = \dfrac{-1}{m_1} = -2$

    The required line passes through the point $P \left(4, -1\right)$

    $\therefore \;$ Equation of the required line is

    $y + 1 = -2 \left(x - 4\right)$

    i.e. $\;$ $y + 1 = -2x + 8$

    i.e. $\;$ $2x + y = 7$

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