Question 1183865
.
Find the equation of the locus of a point which moves so that its distance 
from the point (1,-1) is three times its distance from the line y = 3.
~~~~~~~~~~~~~~~



            The setup equation is  INCORRECT  in the post by @josgaritmetic.


            THEREFORE,  his  "solution"  is  WRONG  from the very first line to very last line.


            For your safety,  ignore his post.


            I came to bring a correct solution.



<pre>
Write the equation for distances, as you read the problem

    {{{sqrt((x-1)^2 + (y- (-1))^2)}}} = 3*|y-3|, 

or

    {{{sqrt((x-1)^2 + (y+1)^2)}}} = 3*|y-3|.


Square both sides

    (x-1)^2 + (y+1)^2 = 9*(y-3)^2.


Simplify and reduce to the standard conic section equation

    (x-1)^2 +  y^2 + 2y + 1 = 9y^2 - 54y + 81

    (x-1)^2 - 8y^2 + 56y + 1  = 81

    (x-1)^2 - 8(y^2 - 7y) + 1 = 81

    (x-1)^2 - 8(y^2 - 2*3.5y + 3.5^2) + 8*3.5^2 + 1 = 81

    (x-1)^2 - 8(y-3.5)^2 = 81 - 8*3.5^2 - 1

    (x-1)^2 - 8(y-3.5)^2 = - 18

    8(y-3.5)^2 - (x-1)^2 = 18


    {{{(y-3.5)^2/((9/4))}}} - {{{(x-1)^2/18}}} = 1.


The last equation describes a hyperbola with the center at the point (1,3.5),

vertical real semi-axis of the length  {{{sqrt(9/4)}}} = {{{3/2}}}  (tranverse semi-axis),

and horizontal imaginary semi-axis of the length  {{{sqrt(18)}}} = {{{3*sqrt(2)}}}.


The hyperbola is open vertically up and down.
</pre>

Solved.


----------------


For hyperbola, its canonical equation, standard form equation, general equation, elements and properties,

see the lessons


&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Quadratic-relations-and-conic-sections/Hyperbola-definition--canonical-equation--characteristic-points-and-elements.lesson>Hyperbola definition, canonical equation, characteristic points and elements</A> 

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Quadratic-relations-and-conic-sections/Hyperbola-focal-property.lesson>Hyperbola focal property</A> 

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Quadratic-relations-and-conic-sections/Tangent-lines-and-normal-vectors-to-a-hyperbola.lesson>Tangent lines and normal vectors to a hyperbola</A> 

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Quadratic-relations-and-conic-sections/Optical-property-of-a-hyperbola.lesson>Optical property of a hyperbola</A> 


&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Quadratic-relations-and-conic-sections/Standard-equation-of-a-hyperbola.lesson>Standard equation of a hyperbola</A> 

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Quadratic-relations-and-conic-sections/Identify-elements-of-a-hyperbola-given-by-its-standard-eqn-NEW.lesson>Identify elements of hyperbola given by its standard equation</A> 

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Quadratic-relations-and-conic-sections/Find-the-standard-equation-of-a-hyperbola-given-by-its-elements.lesson>Find the standard equation of a hyperbola given by its elements</A> 


&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Quadratic-relations-and-conic-sections/General-equation-of-a-hyperbola.lesson>General equation of a hyperbola</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Quadratic-relations-and-conic-sections/Transform-general-eqn-of-a-hyperbola-to-the-standard-form-by-completing-the-square.lesson>Transform general equation of a hyperbola to the standard form by completing the square</A> 

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Quadratic-relations-and-conic-sections/Identify-elements-of-a-hyperbola-given-by-its-gen-eqn.lesson>Identify elements of a hyperbola given by its general equation</A> 


&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Quadratic-relations-and-conic-sections/REVIEW-of-lessons-on-hyperbolas.lesson>OVERVIEW of lessons on hyperbolas</A> 



Also, &nbsp;you have this free of charge online textbook in ALGEBRA-II in this site

&nbsp;&nbsp;&nbsp;&nbsp;<A HREF=https://www.algebra.com/algebra/homework/complex/ALGEBRA-II-YOUR-ONLINE-TEXTBOOK.lesson>ALGEBRA-II - YOUR ONLINE TEXTBOOK</A>.


The referred lesson is the part of this online textbook under the topic 
"<U>Conic sections: Hyperbolas. Definition, major elements and properties. Solved problems</U>".



Save the link to this textbook together with its description


Free of charge online textbook in ALGEBRA-II
https://www.algebra.com/algebra/homework/complex/ALGEBRA-II-YOUR-ONLINE-TEXTBOOK.lesson


into your archive and use when it is needed.



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After seeing my post, &nbsp;@josgarithmetic rewrote his setup exactly as my, &nbsp;so now his post is safe &nbsp;(although useless).


At least, &nbsp;I forced him to make it safe . . .