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Q.
Obtain the equation of the line passing through the intersection of the lines $2x - 3y + 4 = 0$ and $3x + 4y = 5$, and drawn parallel to $y$-axis.
Straight Lines
Solution:
Any line through the intersection of $2x - 3y + 4 = 0$
and $3x + 4y - 5 = 0$ can be taken as
$2x - 3y + 4 + k(3x + 4y - 5) = 0$
or $x(2 + 3k) + y (4k - 3) + 4 - 5k = 0\quad \ldots(i)$
This line is parallel to $y$-axis if $4k - 3 = 0$ i.e., if $k=\frac{3}{4}$.
Substituting $k=\frac{3}{4}$ in $\left(i\right)$, we get
$x\left(2+\frac{9}{4}\right)+y\left(3-3\right)+4-\frac{15}{4}=0$,
i.e., $17x+1=0$