Let #triangle ABC# be the triangle with corners at
#A(6,3) ,B(4,5) and C(2,9)#
Let #bar(AL) , bar(BM) and bar(CN) # be the altitudes of sides
#bar(BC) ,bar(AC) ,and bar(AB)# respectively.
Let #(x,y)# be the intersection of three altitudes .
Slope of #bar(AB) =(5-3)/(4-6)=-1#
#bar(AB)_|_bar(CN)=>#slope of # bar(CN)=1# , # bar(CN)# passes through #C(2,9)#
#:.#The equn. of #bar(CN)# is #:y-9=1(x-2)#
#i.e. color(red)(x-y=-7.....to (1)#
Slope of #bar(BC) =(9-5)/(2-4)=-2#
#bar(AL)_|_bar(BC)=>#slope of # bar(AL)=1/2# , # bar(AL)# passes through #A(6,3)#
#:.#The equn. of #bar(AL)# is #:y-3=1/2(x-6)=>2y-6=x-6#
#i.e. color(red)(x=2y.....to (2)#
Subst. #x=2y# into #(1)# ,we get
#2y-y=-7=>color(blue)( y=-7#
From equn.#(2)# we get
#x=2y=2(-7)=>color(blue)(x=-14#
Hence, the orthocenter of triangle is #(-14,-7)#
