Question

The set of lines $ax+ by+ c = 0$, where $3a + 2b + 4c = 0$ is concurrent at the point... .

Answer 1

The set of lines $ax + by + c = 0$, where $3a + 25 + 4c = 0$ or $\frac {3}{4}a+ \frac {1}{2}b+c=0 $ are concurrent at $\bigg (x=\frac {3}{4},y=\frac {1}{2} \bigg ) i.e.$
comparing the coefficients of $x$ and $y$.
Thus, point of concurrency is $\bigg (\frac {3}{4}, \frac {1}{2} \bigg ).$
Alternate Solution
As, $ax+ by + c = 0$, satisfy $3a + 26 + 4c = 0$ which represents system of concurrent lines whose point of concurrency could be obtained by comparison as,
$ax+by+c \equiv \frac {3a}{4}+ \frac {2}{4}b+c $
$\Rightarrow \, x= \frac {3}{4}, \, y= \frac {1}{2} $ is point of concurrency.
$\therefore \, \bigg (\frac {3}{4}, \frac {1}{2} \bigg ) $ is the required point.