SOLUTION: A point P(x,y) moves in such a way that its distance from (3,2) is always one half of its distance from (-1,3). find the equation of the locus

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Question 1191709: A point P(x,y) moves in such a way that its distance from (3,2) is always one half of its distance from (-1,3). find the equation of the locus
Found 2 solutions by josgarithmetic, ikleyn:
Answer by josgarithmetic(39617) About Me  (Show Source):
Answer by ikleyn(52786) About Me  (Show Source):
You can put this solution on YOUR website!
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A point P(x,y) moves in such a way that its distance from (3,2) is always
one half of its distance from (-1,3). find the equation of the locus
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Write an equation as you read the problem

    sqrt%28%28x-3%29%5E2%2B%28y-2%29%5E2%29 = %281%2F2%29%2Asqrt%28%28x%2B1%29%5E2+%2B+%28y-3%29%5E2%29.


Square both side

    (x-3)^2 + (y-2)^2 = %281%2F4%29.( (x+1)^2 + (y-3)^2 )


Simplify it step by step

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

    4*(x^2 - 6x + 9 + y^2 -4y + 4) = x^2 + 2x + 1 + y^2 - 6y + 9

    4x^2 - 24x + 36 + 4y^2 - 16y + 16 = x^2 + 2x + 1 + y^2 - 6y + 9


The further simplification is routine bothering calculations, from which you will learn NOTHING. 

It is a standard completing the square procedure.


So I will complete by referring to specialized online calculator

https://www.equationcalc.com/conics-section-calculator


Copy/paste the last equation to this calculator input port and get there the answer: final equation is 


    %28x-%2813%2F3%29%29%5E2 + %28y-%285%2F3%29%29%5E2 = %282%2F3%29%5E2%2A17.


It is the standard form equation of the circle with the center at the point (13%2F3,5%2F3)  with the radius of  %282%2F3%29%2Asqrt%2817%29.

Solved.


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It is really BAD problem to learn from.

My condolences . . .


Would somebody asked me about an educational value of this problem, I'd say it is zero.