SOLUTION: find the equation of the set of points whose distance of each element from (3,2) is twice its distance from (-1,4). thanks youuu

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Question 1130575: find the equation of the set of points whose distance of each element from (3,2) is twice its distance from (-1,4). thanks youuu
Found 2 solutions by josgarithmetic, ikleyn:
Answer by josgarithmetic(39630) About Me  (Show Source):
You can put this solution on YOUR website!
Applying the distance formula
cross%28%28x-3%29%5E2%2B%28y-2%29%5E2=2%28x%2B1%29%5E2%2B2%28y-4%29%5E2%29
sqrt%28%28x-3%29%5E2%2B%28y-2%29%5E2%29=2%2Asqrt%28%28x%2B1%29%5E2%2B%28y-4%29%5E2%29
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3x%5E2%2B14x%2B3y%5E2-28y%2B55=0
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completing the squares and simplifying,
%28x%2B%287%2F3%29%29%5E2%2B%28y-14%2F3%29%5E2=80%2F9

Answer by ikleyn(52879) About Me  (Show Source):
You can put this solution on YOUR website!
.
find the equation of the set of points whose distance of each element from (3,2) is twice its distance from (-1,4). highlight%28cross%28thanks%29%29 thank youuu
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Let (x,y) is the point of the locus (of the set of points).


Its distance from the point (3,2) is  sqrt%28%28x-3%29%5E2%2B%28y-2%29%5E2%29.

Its distance from the point (-1,4) is  sqrt%28%28x-%28-1%29%29%5E2%2B%28y-4%29%5E2%29 = sqrt%28%28x%2B1%29%5E2%2B%28y-4%29%5E2%29.


From the condition,

    sqrt%28%28x-3%29%5E2%2B%28y-2%29%5E2%29 = 2%2Asqrt%28%28x%2B1%29%5E2%2B%28y-4%29%5E2%29.


Square both sides

    (x-3)^2 + (y-2)^2 = 4*((x+1)^2 + (y-4)^2).


FOIL

    x^2 - 6x + 9 + y^2 - 4y + 4 = 4x^2 + 8x + 4 + 4y^2 - 32y + 64

    3x^2 + 14x + 3y^2 - 28y = -51

    x^2 + 14%2F3%29%2Ax + y^2 - %2828%2F3%29%2Ay = - 17

    x^2 + %2814%2F3%29%2Ax + %2814%2F6%29%5E2 + y^2 - %2828%2F3%29%2Ay + %2828%2F6%29%5E2 = -17 + %2814%2F6%29%5E2 + %2828%2F6%29%5E2

    (x+14/6)^2 + (y-28/6)^2 = -17 + %2814%5E2%2B28%5E2%29%2F6%5E2 = -17 + 980%2F36 = 368%2F36.


Answer.  This locus (= this set of points) is the circle with the equation

         %28x%2B7%2F3%29%5E2 + %28y-14%2F3%29%5E2 = 368%2F36.

         The circle is centered at (-7%2F3,14%2F3)  and has the radius  of  sqrt%28368%29%2F6  units.

Solved.

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Be aware :   The equation   %28x-3%29%5E2%2B%28y-2%29%5E2=2%28x%2B1%29%5E2%2B2%28y-4%29%5E2   which @josgarithmetic proposes you to simplify,  is INCORRECT  (!)