Equations of the lines through the point (3, 2) and making an angle of 45° with the line x – 2y = 3 are ____.

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Vishal Baghel | Contributor-Level 10

a year ago

This is a Fill in the Blank Type Questions as classified in NCERT Exemplar

Sol:

$\begin{array}{l} G i v e n l i n e i s x - 2 y = 3 a n d t h e i s (3, 2) \\ E q u a t i o n o f a l i n e t h r o u g h t h e (3, 2) i s \\ A n g l e b e t w e e n e q n . (i) a n d t h e g i v e n l i n e x - 2 y = 3 w h o s e s l o p e i s \frac{1}{2} \\ ∴ t a n θ = | \frac{m_{1} - m_{2}}{1 + m_{1} m_{2}} | \\ \Rightarrow t a n 4 5^{0} = | \frac{m - \frac{1}{2}}{1 + m * \frac{1}{2}} | \\ \Rightarrow 1 = | \frac{m - \frac{1}{2}}{1 + \frac{m}{2}} | \Rightarrow \frac{m - \frac{1}{2}}{1 + \frac{m}{2}} = \pm 1 \\ T a k i n g (+) s i g n \\ \Rightarrow \frac{m - \frac{1}{2}}{1 + \frac{m}{2}} = 1 \Rightarrow m - \frac{1}{2} = 1 + \frac{m}{2} \\ \Rightarrow m - \frac{m}{2} = 1 + \frac{1}{2} \Rightarrow \frac{m}{2} = \frac{3}{2} \Rightarrow m = 3 \\ T a k i n g (-) s i g n \\ \Rightarrow \frac{m - \frac{1}{2}}{1 + \frac{m}{2}} = - 1 \Rightarrow m - \frac{1}{2} = - 1 - \frac{m}{2} \\ \Rightarrow m + \frac{m}{2} = - 1 + \frac{1}{2} \Rightarrow \frac{3 m}{2} = \frac{1}{2} \Rightarrow m = - \frac{1}{3} \\ S o, t h e r e q u i r e d e q u a t i o n s a r e, \\ W h e n m = 3, \\ y - 2 = 3 (x - 3) \\ \Rightarrow y - 2 = 3 x - 9 \\ \Rightarrow 3 x - y - 7 = 0 \\ W h e n m = - \frac{1}{3}, \\ y - 2 = - \frac{1}{3} (x - 3) \\ \Rightarrow 3 y - 6 = - x + 3 \\ \Rightarrow x + 3 y - 9 = 0 \\ H e n c e, t h e v a l u e o f t h e f i l l e r a r e 3 x - y - 7 = 0 a n d x + 3 y - 9 = 0 . \end{array}$

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