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Find the coordinate(s) of the point(s) that satisfy the following conditions:

  • equidistant from the points $(4,-7)$ and $(8,5)$
  • distance of 5 from the point $(-5,1)$
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1 Answer 1

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Hint: The distance from a point $p_1=(x_1,y_1)$ to another point $p_2(x_2,y_2)$ is given by $$d\left(p_1,p_2\right) = \sqrt{\left(x_1-x_2\right)^2+\left(y_1-y_2\right)^2}$$

Now you are searching a special point $p=(x,y)$ that fullfills

  1. $d\left(p,(4,-7)\right) = d\left(p,(8,5)\right)$
  2. $d\left(p,(-5,1)\right) = 5$

Now set up the resulting equations and solve the non-linear system for $x$ and $y$

  1. $(x-4)^2 + (y+7)^2 = (x-8)^2+(y-5)^2$
  2. $(x+5)^2 + (y-1)^2 = 5^2$

Note that $x^2$ and $y^2$ cancels out in the first equation, so it is easy to solve this either to $x$ or $y$ and put it in the second equation.

Can you go ahead?

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