# 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

$B \left(x , y\right) = \left(\frac{25}{17} , \frac{134}{17}\right) = \left(1.47 , 7.88\right)$

#### Explanation:

By Tangent-Secant theorem, we know that,
$P {C}^{2} = P A \cdot P B$
$P {C}^{2} = {\left(4 - 14\right)}^{2} + {\left(1 + 10\right)}^{2} = 100 + 121 = 221$,
$P A = \sqrt{{\left(6 - 14\right)}^{2} + {\left(5 + 10\right)}^{2}} = \sqrt{64 + 225} = \sqrt{289} = 17$,
$\implies P B = \frac{P {C}^{2}}{P A} = \frac{221}{17} = 13$
$\implies P B : B A = 13 : \left(17 - 13\right) = 13 : 4$
$\implies B$ divides line $P A$ in the ratio of $13 : 4$
Section formula : if a point $B \left(x , y\right)$ divides a line joining $P \left({x}_{1} , {y}_{1}\right) \mathmr{and} A \left({x}_{2} , {y}_{2}\right)$ in the ratio of $m : n$,
then $B \left(x , y\right) = \left(\frac{m \cdot {x}_{2} + n \cdot {x}_{1}}{m + n} , \frac{m \cdot {y}_{2} + n \cdot {y}_{1}}{m + n}\right)$

$\implies B \left(x , y\right) = \left(\frac{13 \cdot 5 + 4 \cdot \left(- 10\right)}{13 + 4} , \frac{13 \cdot 6 + 4 \cdot 14}{13 + 4}\right)$
$= \left(\frac{25}{17} , \frac{134}{17}\right) = \left(1.47 , 7.88\right)$

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