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Question
Show that the tangent of an angle between the lines \[\frac{x}{a} + \frac{y}{b} = 1 \text { and } \frac{x}{a} - \frac{y}{b} = 1\text { is } \frac{2ab}{a^2 - b^2}\].
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Solution
We have
\[\frac{x}{a} + \frac{y}{b} = 1 \text { and } \frac{x}{a} - \frac{y}{b} = 1\]
\[ \Rightarrow \frac{y}{b} = - \frac{x}{a} + 1 \text { and } \frac{y}{b} = \frac{x}{a} - 1\]
\[ \Rightarrow y = - \frac{b}{a}x + b \text { and} y = \frac{b}{a} - b\]
The slopes of the two lines are \[- \frac{b}{a}\text { and } \frac{b}{a}\]
Now, the tangent of an angle between the lines is given by
\[\frac{\frac{b}{a} + \frac{b}{a}}{1 - \frac{b}{a} \times \frac{b}{a}}\]
\[ = \frac{\frac{2b}{a}}{\frac{a^2 - b^2}{a^2}}\]
\[ = \frac{2ab}{a^2 - b^2}\]
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