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Given:

Line a²x + ay + 1 = 0 is perpendicular to the line x – ay = 1

To prove:

The line a²x + ay + 1 = 0 is perpendicular to the line x – ay = 1 for all non-zero real values of a.

Concept Used:

Product of slope of perpendicular line is -1.

Explanation:

The given lines are

a²x + ay + 1 = 0 … (1)

x − ay = 1 … (2)

Let m₁ and m₂ be the slopes of the lines (1) and (2).

m₁m₂ = -

Hence proved, line a²x + ay + 1 = 0 is perpendicular to the line x − ay = 1 for all non-zero real values of a.