Three points have coordinates #A(1,2)#, #B(9,0)# and #C(6,t)#. Calculate the value(s) of t if : (a) #angleABC=90^@# (b) AC is perpendicular to BC?

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Answer 1 · 2018-02-06T16:30:01

(a) #t=-12# and (b) #t=5# or #t=-3#

Slope of line joining #(x_1,y_1)# and #(x_2,y_2)# is #(y_2-y_1)/(x_2-x_1)#

Therefore slope of #AB# is #(0-2)/(9-1)=-2/8=-1/4#,

slope of #BC# is #(t-0)/(6-9)=-t/3# and

slope of #AC# is #(t-2)/(6-1)=(t-2)/5#.

Further if two lines are perpendicular, product of their slopes is #-1#

Hence (a) if #/_ABC=90^@#, we have #AB# perpendicular to #BC# i.e. #-1/4xx(-t/3)=-1#

or #t/12=-1# i.e. #t=-12#

(b) if #AC# is perpendicular to #BC#, then

#(t-2)/5xx(-t/3)=-1# i.e. #t(t-2)=15#

or #t^2-2t-15=0# or #(t-5)(t+3)=0# i.e. #t=5# or #t=-3#