A straight line 4x + 3y = 24 forms a triangle with the coordinate axes. What is the distance between the orthocentre and circumcentre of the triangle so formed?
Consider the diagram below where the line 4x + 3y = 24 forms a triangle with the coordinate axes.
We see that it forms a right triangle with sides 6, 8 and 10. And in a right triangle, the orthocenter lies at the vertex where the right angle is formed. Thus, orthocenter lies at (0,0). The circumcenter in a right triangle is at the mid-point of the hypotenuse. Thus, mid-point of (6,0) and (0,8) is (3,4). Thus, the distance between (0,0) and (3,4) is 5 units.
If you think about it the other way, the line joining the orthocenter and the circumcenter is nothing but the median to the hypotenuse. And we also know that the median to the hypotenuse is the circumradius too. And in a right triangle, the circumradius is half of the hypotenuse. Hence 5 units.