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.

Answer 1

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 = - \(\frac{a^2}{a}\times\frac{1}{a}\) = -1

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