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Q.
Find the equation of the line passing through the point $(-4,5)$ and the point of intersection of the lines $4x - 3y + 7 = 0$ and $2x + 3 y + 5 = 0$.
Straight Lines
Solution:
The given lines are
$4x - 3y + 7 = 0\quad \ldots(i)$
and $2x + 3y + 5 = 0\quad \ldots(ii)$
The equation of the family of lines passing through the intersection of the lines $(i)$ and $(ii)$ is
$(4x -3y + 7) + k(2x + 3y + 5) = 0\quad \ldots(iii)$
where $k$ is a parameter.
Since, it passes through $(-4,5)$, we have
$4 \times (-4) - 3 \times 5 + 7 + k\{2 \times (-4) + 3 \times 5 +5\} = 0$
$\Rightarrow - 16 - 15 + 7 + k(- 8 + 15 + 5) = 0$
$\Rightarrow -24 + 12k =0$
$\Rightarrow k=2$
Substituting this value of $k$ in $(iii)$, the equation of the required line is
$(4x - 3y + 7) + 2(2x + 3y + 5) = 0$
$\Rightarrow 8x + 3y + 17 = 0$