+0

# the coordinates of Q

+3
410
3

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

Feb 17, 2015

#1
+27377
+10

.

Feb 17, 2015

#1
+27377
+10

.

Alan Feb 17, 2015
#2
+94558
+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

Feb 17, 2015
#3
+27377
0

Thanks Chris.  I was a little lazy here and just got Mathcad to solve the equations for me!

Feb 18, 2015