Question

Find the equation of a straight line passing through the point of intersection of x + 2y + 3 = 0 and 3x + 4y + 7 = 0 and perpendicular to the straight line x − y + 9 =0

Answer 1

The equation of the straight line passing through the points of intersection of x + 2y + 3 = 0 and 3x + 4y + 7 = 0 is

x + 2y + 3 + λ(3x + 4y + 7) = 0

$\Rightarrow$(1 + 3λ)x + (2 + 4λ)y + 3 + 7λ = 0

$\Rightarrow y=-\left(\frac{1+3\lambda }{2+4\lambda }\right)x-\left(\frac{3+7\lambda }{2+4\lambda }\right)$

The required line is perpendicular to x − y + 9 = 0 or, y = x + 9

$$\begin{gathered} \therefore \left(-\frac{1+3\lambda }{2+4\lambda }\right)\times 1=-1 \\ \Rightarrow \lambda =-1 \end{gathered}$$

Required equation is given below:

(1 − 3)x + (2 − 4)y + 3 − 7 = 0

$\Rightarrow$ x + y + 2 = 0