Straight Line and Pair of Straight Lines 1 Question 57
57. Find the equations of the line passing through the point $(2,3)$ and making intercept of length 3 unit between the lines $y+2 x=2$ and $y+2 x=5$.
(1991, 4M)
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Solution:
- Let $l$ makes an angle $\alpha$ with the given parallel lines and intercept $A B$ is of 3 units.
Now, distance between parallel lines
$$ \begin{aligned} & =\frac{|5-2|}{\sqrt{1^{2}+2^{2}}}=\frac{3}{\sqrt{5}} \\ \therefore \quad \sin \alpha & =\frac{1}{\sqrt{5}}, \cos \alpha=\frac{2}{\sqrt{5}} \\ \text {, and } \quad \tan \alpha & =\frac{1}{2} \end{aligned} $$
$\Rightarrow$ Equation of straight line passing through $(2,3)$ and making an angle $\alpha$ with $y+2 x=5$ is
$$ \begin{array}{rlrl} & & \frac{y-3}{x-2} & =\tan (\theta+\alpha) \\ & \text { and } & \frac{y-3}{x-2} & =\frac{\tan \theta+\tan \alpha}{1-\tan \theta \tan \alpha} \\ \Rightarrow & \frac{y-3}{x-2} & =\frac{\tan \theta-\tan \alpha}{1+\tan \theta \tan \alpha} \\ \Rightarrow & & \frac{y-3}{x-2} & =-\frac{3}{4} \text { and } \frac{y-3}{x-2}=\frac{1}{0} \end{array} $$
