Answer :

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).


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