Question #50f9b

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Answer 1 · 2016-02-23T18:42:06

#(a) k->y=(-4x+25)/3, l->y=(-x+10)/2; (b) D(4,3), (x-4)^2+(y-3)^2=50, r=5sqrt(2)#

Repeating the points
#A(5,10)#
#B(-3,4)#
#C(-1,-2)#
So the midpoints of interest are
#M_(AB)(1,7)#
#M_(AC)(2,4)#

(a)
Finding the slopes (#k_n=(Delta y)/(Delta x) and p_n=-1/(k_n)#)

#AB-> k_1=(4-10)/(-3-5)=(-6)/-8=3/4# => #p_1=-4/3#
#AC-> k_2=(-2-10)/(-1-5)=(-12)/(-6)=2# => #p_2=-1/2#

Equations of lines

#k-> (y-7)=(-4/3)(x-1)# => #y=(-4x+4)/3+7# => #y=(-4x+25)/3#
#l->(y-4)=(-1/2)(x-2)# => #y=(-x+2)/2+4# => #y=(-x+10)/2#

(b)
#D=knnl#

Finding D

#(-4x+25)/3=(-x+10)/2# => #-8x+50=-3x+30# => #5x=20# => #x=4#
#->y=(-4+10)/2=6/2# => #y=3#
#-> D(4,3)#

Since D is the circumcenter then
#AD=BD=CD=r#
(choosing CD)# => r=sqrt((4+1)^2+(3+2)^2)=sqrt(25+25)=sqrt(50)=5sqrt(2)#

Circle's equation
#(x-x_D)^2+(y-y_D)^2=r^2#
#(x-4)^2+(y-3)^2=50#