A line passing through the points (a, 2a) and (-2, 3) is perpendicular to the line $4 x + 3 y + 5 = 0$ , find the value of a.

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Solution

Slope of line through the points

$(a, 2 a) (x_{1}, y_{1}), (- 2, 3) (x_{2}, y_{2})$

$\Rightarrow m_{1} = \frac{y_{2} - y_{1}}{x_{2} - x_{1}} = \frac{3 - 2 a}{- 2 - a}$

Also, slope of line $x - a y = 1$ in the form

$y = m x + c$

$4 x + 3 y + 5 = 0$

$y = \frac{- 4}{3} x - \frac{5}{3}$

$\Rightarrow m_{2} = \frac{- 4}{3}$

If two lines are perpendicular then,

$m_{1} m_{2} = - 1$

$(\frac{3 - 2 a}{- 2 - a}) (\frac{- 4}{3}) = - 1$

$- 12 + 8 a = 6 + 3 a$

$5 a = 18$

$a = \frac{18}{5}$

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