Question

The equation of the line with slope −3/2 and which is concurrent with the lines 4x + 3y − 7 = 0 and 8x + 5y − 1 = 0 is
(a) 3x + 2y − 63 = 0
(b) 3x + 2y − 2 = 0
(c) 2y − 3x − 2 = 0
(d) none of these

Answer 1

(b) 3x + 2y $-$2=0

Given:

4x + 3y − 7 = 0 ... (1)

8x + 5y − 1 = 0 ... (2)

The equation of the line with slope $-\frac{3}{2}$ is given below:

$y=-\frac{3}{2}x+c$

$\Rightarrow \frac{3}{2}x+y-c=0$ ... (3)

The lines (1), (2) and (3) are concurrent.

$$\begin{gathered} \therefore \left|\begin{array}{ccc}4& 3& -7\\ 8& 5& -1\\ \frac{3}{2}& 1& -c\end{array}\right|=0 \\ \Rightarrow 4\left(-5c+1\right)-3\left(-8c+\frac{3}{2}\right)-7\left(8-\frac{15}{2}\right)=0 \\ \Rightarrow -20c+4+24c-\frac{9}{2}-56+\frac{105}{2}=0 \\ \Rightarrow \frac{-40c+8+48c-9-112+105}{2}=0 \\ \Rightarrow 8c=8 \\ \Rightarrow c=1 \end{gathered}$$

On substituting c = 1 in $y=-\frac{3}{2}x+c$, we get:

$$\begin{gathered} y=-\frac{3}{2}x+1 \\ \Rightarrow 3x+2y-2=0 \end{gathered}$$