Question

Two lines are drawn through the point (1, 3) one of slope and the other of slope −2. Write the coordinates of one more point on each of these lines. Prove that these lines are perpendicular to each other.

Answer 1

Let the given point be A (1, 3).

Let the point B (x, y) lie on the line passing through point A.

Given that the slope of the first line passing through point A is

⇒

⇒ 2y − 6 = x − 1

⇒ x − 2y + 5 = 0

This is the equation of the line passing through point A and having slope

Any point on this line can be found out by assuming x to have any value, say 2 and putting it in the above equation.

2 − 2y + 5 = 0

⇒ −2y + 7 = 0

⇒ y =

Thus, the coordinates of one more point on the first line are

Also, given that the slope of the second line passing through point A is −2.

⇒

⇒ y − 3 = −2x + 2

⇒ 2x + y − 5 = 0

This is the equation of the line passing through point A and having slope −2.

Any point on this line can be found out by assuming x to have any value, say 2 and putting it in the above equation.

4 + y − 5 = 0

⇒ y − 1 = 0

⇒ y = 1

Thus, the coordinates of one more point on the second line are (2, 1).

Let m₁ = , m₂ = −2

For two lines to be perpendicular, the product of their slopes should be equal to −1.

m₁ m₂ = × (−2) = −1

Therefore, the given lines are perpendicular to each other.