Find the equations of the lines through the point of intersection of the lines x – y + 1 = 0 and 2x – 3y + 5 = 0 and whose distance from the point (3, 2) is √5.

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Vishal Baghel | Contributor-Level 10

a year ago

This is a Long Answer Type Questions as classified in NCERT Exemplar

Sol:

$\begin{array}{l} G i v e n e q u a t i o n s a r e \\ x - y + 1 = 0 \dots (i) \\ a n d 2 x - 3 y + 5 = 0 \dots (i i) \\ S o l v i n g e q n . (i) a n d e q n . (i i) w e g e t \\ 2 x - 2 y + 2 = 0 \\ 2 x - 3 y + 5 = 0 \\ \underline{(-) (+) (-)} \\ y - 3 = 0 ∴ y = 3 \\ F r o m e q n . (i) w e h a v e \\ x - 3 + 1 = 0 \Rightarrow x = 2 \\ S o, (2, 3) i s t h e o f o f e q n . (i) a n d e q n . (i i) . \\ L e t m b e t h e s l o p e o f t h e r e q u i r e d l i n e \\ ∴ E q u a t i o n o f t h e l i n e i s \\ y - 3 = m (x - 2) \\ \Rightarrow y - 3 = m x - 2 m \\ \Rightarrow m x - y + 3 - 2 m = 0 \\ Since, t h e p e r p e n d i c u l a r f r o m (3, 2) t o t h e l i n e i s \frac{7}{5} t h e n \\ \frac{7}{5} = | \frac{m (3) - 2 + 3 - 2 m}{\sqrt[]{m^{2} + 1}} | \\ \Rightarrow \frac{49}{25} = \frac{{(3 m - 2 + 3 - 2 m)}^{2}}{m^{2} + 1} \\ \Rightarrow \frac{49}{25} = \frac{{(m + 1)}^{2}}{m^{2} + 1} \\ \Rightarrow 49 m^{2} + 49 = 25 m^{2} + 50 m + 25 \\ \Rightarrow 49 m^{2} - 25 m^{2} - 50 m + 49 - 25 = 0 \\ \Rightarrow 24 m^{2} - 50 m + 24 = 0 \\ \Rightarrow 12 m^{2} - 25 m + 12 = 0 \\ \Rightarrow 12 m^{2} - 16 m - 9 m + 12 = 0 \\ \Rightarrow 4 m (3 m - 4) - 3 (3 m - 4) = 0 \\ \Rightarrow (3 m - 4) (4 m - 3) = 0 \\ \Rightarrow 3 m - 4 = 0 a n d 4 m - 3 = 0 \\ \Rightarrow m = \frac{4}{3}, \frac{3}{4} \\ E q u a t i o n o f t h e l i n e t a k i n g m = \frac{4}{3} i s \\ y - 3 = \frac{4}{3} (x - 2) \\ \Rightarrow 3 y - 9 = 4 x - 8 \Rightarrow 4 x - 3 y + 1 = 0 \\ E q u a t i o n o f t h e l i n e t a k i n g m = \frac{3}{4} i s \\ y - 3 = \frac{3}{4} (x - 2) \\ \Rightarrow 4 y - 12 = 3 x - 6 \Rightarrow 3 x - 4 y + 6 = 0 \\ H e n c e, t h e r e q u i r e d e q u a t i o n s a r e 4 x - 3 y + 1 = 0 a n d 3 x - 4 y + 6 = 0 . \end{array}$

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