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Question
Show that the line a2x + ay + 1 = 0 is perpendicular to the line x − ay = 1 for all non-zero real values of a.
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Solution
The given lines are
a2x + ay + 1 = 0 ... (1)
x − ay = 1 ... (2)
Let \[m_1 \text { and } m_2\] be the slopes of the lines (1) and (2).
\[m_1 m_2 = - \frac{a^2}{a} \times \frac{1}{a}\]
\[ = - 1\]
Hence, line a2x + ay + 1 = 0 is perpendicular to the line x − ay = 1 for all non-zero real values of a.
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