Answer to Question #90264 in Analytic Geometry for Juan .D

Question #90264
The coordinates of points A and B are A (16,4) and B (34,40). What are the coordinates of a point that partitions segment AB in a ratio of 1:5?
1
Expert's answer
2019-05-27T13:08:53-0400

Let's find the length of AB:


(1) "AB=\\sqrt{\\smash[b]{(x_2-x_1)}^{\\smash{2}}+{(y_2-y_1)}^{\\smash{2}}}"


"AB=\\sqrt{\\smash[b]{(34-16)}^{\\smash{2}}+{(40-4)}^{\\smash{2}}}"


"AB=\\sqrt{\\smash[b]324+1296}=\\sqrt{\\smash[b]1620}=18*\\sqrt{\\smash[b]5}"


Let point L (xL;yL) will be a point that partition segment AB in a ratio of 1:5. It means that:


"AL=AB*\\frac{1}{6}" and "BL=AB*\\frac{5}{6}"


Lengths of segments AL and BL are respectively equal:


(2) "AL=\\sqrt{\\smash[b]{(x_A-x_L)}^{\\smash{2}}+{(y_A-y_L)}^{\\smash{2}}}=18*\\sqrt{\\smash[b]{(5)}}\/6=3*\\sqrt{\\smash[b]{(5)}}"


(3) "BL=\\sqrt{\\smash[b]{(x_B-x_L)}^{\\smash{2}}+{(y_B-y_L)}^{\\smash{2}}}=18*\\sqrt{\\smash[b]{(5)}}*5\/6=15*\\sqrt{\\smash[b]{(5)}}"


We now coordinates of points A (16;4) and B (34;40):


"\\begin{cases}\n\\sqrt{\\smash[b]{(16-x_L)}^{\\smash{2}}+{(4-y_L)}^{\\smash{2}}}=3*\\sqrt{\\smash[b]{(5)}} \\\\\n\\sqrt{\\smash[b]{(34-x_L)}^{\\smash{2}}+{(40-y_L)}^{\\smash{2}}}=15*\\sqrt{\\smash[b]{(5)}}\n\\end{cases}"


"\\begin{cases}\n(16-x_L)^{\\smash{2}}+(4-y_L)^{\\smash{2}}=45 \\\\\n(34-x_L)^{\\smash{2}}+(40-y_L)^{\\smash{2}}=1125\n\\end{cases}"


"\\begin{cases}\n256-32*x_L+x_L^2+16-8*y_L+y_L^2=45 \\\\\n1156-68*x_L+x_L^2+1600-80*y_L+y_L^2=1125\n\\end{cases}"


Let's subtract the first equation from the second:


"900-36*x_L+1584-72*y_L=1080"


"39-x_L-2*y_L=0"


(4) "x_L=39-2*y_L"


Substitute value of xL in equation (2):


"\\sqrt{\\smash[b]{(16-39+2*y_L)}^{\\smash{2}}+{(4-y_L)}^{\\smash{2}}}=3*\\sqrt{\\smash[b]{(5)}}"


"(2*y_L-23)^{\\smash{2}}+(4-y_L)^{\\smash{2}}=45"


"4*y_L^2-92*y_L+529+16-8*y_L+y_L^2-45=0"


"500-100*y_L+5*y_L^2=0"


"100-20*y_L+*y_L^2=0"


"(10-y_L)^2=0\\implies y_L=10"


Substitute value of yL in equation (4):


"x_L=39-2*10=19"


Coordinates of point that partitions segment AB in a ratio of 1:5 is (19;10)


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