Given:

Lines x² – y² = 0 and x + 6y = 18.

To find:

The coordinates of the orthocenter of the triangle formed by the lines x² – y² = 0 and x + 6y = 18.

Explanation:

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 × 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).