The line 2x+y=4 intersects X and Y axes at points A and B respectively. Let C(p, q) be the point such that the lines AC and BC are perpendicular. The distance of point C from the midpoint of segment AB is...

Step-by-step solution

The line 2x+y=4 intersects the X-axis at A(2, 0) and the Y-axis at B(0, 4) . The midpoint of segment AB is M ( 2+0 2 , 0+4 2 ) = (1, 2) . Since the lines AC and BC are perpendicular, ACB = 90^ . This implies that the point C lies on a circle with AB as its diameter. The distance of point C from the midpoint of segment AB is the radius of this circle. The length of the diameter AB is (2-0)^2 + (0-4)^2 = 4+16 = 20 = 2 5 . The radius of the circle is AB 2 = 2 5 2 = 5 . Thus, the distance of point C from the midpoint o