Find the equations of the lines through the point of intersection of the lines x − y + 1 = 0 and 2x − 3y+ 5 = 0, whose distance from the point(3, 2) is 7/5.
Solution
The equations of the lines through the point of intersection of the lines x − y + 1 = 0 and 2x − 3y + 5 = 0 is given by
x − y + 1 + a(2x − 3y + 5) = 0
⇒ (1 + 2a)x + y(−3a − 1) + 5a + 1 = 0 .....(1)
The distance of the above line from the point is given by \[\frac{3\left( 2a + 1 \right) + 2\left( - 3a - 1 \right) + 5a + 1}{\sqrt{\left( 2a + 1 \right)^2 + \left( - 3a - 1 \right)^2}}\]
\[\therefore \frac{\left| 3\left( 2a + 1 \right) + 2\left( - 3a - 1 \right) + 5a + 1 \right|}{\sqrt{\left( 2a + 1 \right)^2 + \left( - 3a - 1 \right)^2}} = \frac{7}{5}\]
\[ \Rightarrow \frac{\left| 5a + 2 \right|}{\sqrt{13 a^2 + 10a + 2}} = \frac{7}{5}\]
\[ \Rightarrow 25 \left( 5a + 2 \right)^2 = 49\left( 13 a^2 + 10a + 2 \right)\]
\[ \Rightarrow 6 a^2 - 5a - 1 = 0\]
\[ \Rightarrow a = 1, - \frac{1}{6}\]
Substituting the value of a in (1), we get
3x − 4y + 6 = 0 and 4x − 3y + 1 = 0
