Question

Orthocenter of the triangle formed by the lines \[x+y=1\] and \[xy=0\] is
(a) (0, 0)
(b) (0, 1)
(c) (1, 0)
(d) (-1, +1)

Answer 1

Hint: We will use the definition of orthocenter to solve this question. We will draw the figure from the given lines in the question to understand where the three altitudes of the three sides intersect.

Complete step-by-step answer:
Before proceeding with the question we should understand the concept of orthocenter.
The orthocenter of the triangle is the intersection of the triangle’s three altitudes. And it is represented by the letter H. The orthocenter is not always inside the triangle. If the triangle is obtuse, it will be outside. In the case of equilateral triangle incenter, circumcenter, centroid and orthocenter occur at the same point.
Now it has been mentioned in the question that the triangle is formed by the lines \[x+y=1\] and \[xy=0\].
So from \[xy=0\] we get x equal to 0 and y equal to 0. So the triangle is bounded by \[x=0\], \[y=0\] and \[x+y=1\].
Drawing the figure from the given details mentioned in the question we get,

So now from the figure we can see that the three altitudes of all the three sides intersect at (0,0). So according to the definition, orthocenter is (0,0).
Hence the correct answer is option (a)

Note: Remembering the definition of orthocenter is very important here. Also to construct the line \[x+y=1\] we first put y equal to 0 in it to get the x coordinate and then we put x equal to 0 in it to get the y coordinate, finally we join the two coordinates and that is the line \[x+y=1\].