Question

Write the coordinates of the orthocentre of the triangle formed by the lines x² − y² = 0 and x + 6y = 18.

Answer 1

The equation x² − y² = 0 represents a pair of straight line, which can be written in the following way:
(x + y)(x − y) = 0

So, the lines can be written separately in the following manner:
x + y = 0          ... (1)
x − y = 0          ... (2)
The third line is
x + 6y = 18      ... (3)
Lines (1) and (2) are perpendicular to each other as their slopes are −1 and 1, respectively
⇒ −1 \[\times\] 1 = −1

Therefore, the triangle formed by the lines (1), (2) and (3) is a right-angled triangle.
Thus, the orthocentre of the triangle formed by the given lines is the intersection of x + y = 0 and x − y = 0, which is (0, 0).