p(4,3) is a point in a rectangular coordinate plane.It is known that Q is a point vertically below P such that the orthocentre of triangle OPQ is H(1,0) m where o is the origin.

ai)the coorinates of Q

ii)Hence, the equation of circle which pass through O,P and Q

Guest Feb 17, 2015

#2**+5 **

Nice, Alan.....I always like these kind of problems....!!!

Here's the solving of the simultaneous equations....using the last two, we have

(x - 4)^2 + (y -3)^2 = (x - 4)^2 + (y + 4)^2 →

(y - 3) ^2 = (y + 4)^2

y^2 - 6y + 9 = y^2 + 8y + 16

14y + 7 = 0 → y = -1/2

And using the first two equations, we have

x^2 + (-1/2)^2 = (x - 4)^2 + (3 +1/2)^2 →

x^2 + 1/4 = x^2 - 8x + 16 + 49/4

8x = 28

x =28 / 8 = 7/2

And using the first equation, we have

(7/2)^2 + (1/2)^2 = r^2

49/4 + 1/4 = r^2

50/4 = r^2

(5√2)/ 2 = r = (5/2)√2

CPhill Feb 17, 2015