Point P(-10,14) is a point external to a circle. Points A(5,6), B(x,y), C(1,4) are on the circle. Line PC is tangent to circle and P, A, and B are collinear, with B between P and A. Find the coordinates of point B. Give your answers two decimal places?

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Dec 16, 2017

Answer:

#B(x,y)=(25/17,134/17)=(1.47,7.88)#

Explanation:

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By Tangent-Secant theorem, we know that,
#PC^2=PA*PB#
#PC^2=(4-14)^2+(1+10)^2=100+121=221#,
#PA=sqrt((6-14)^2+(5+10)^2)=sqrt(64+225)=sqrt289=17#,
#=> PB=(PC^2)/(PA)=221/17=13#
#=> PB:BA=13:(17-13)=13:4#
#=> B# divides line #PA# in the ratio of #13:4#
Section formula : if a point #B(x,y)# divides a line joining #P(x_1,y_1) and A(x_2,y_2)# in the ratio of #m:n#,
then #B(x,y) = ((m*x_2+n*x_1)/(m+n), (m*y_2+n*y_1)/(m+n))#

#=> B(x,y)=((13*5+4*(-10))/(13+4), (13*6+4*14)/(13+4))#
#=(25/17,134/17)=(1.47,7.88)#

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