Wikipedia:Reference desk/Archives/Mathematics/2017 July 11

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July 11

Locus

How do you find the locus of points equidistant from two different points like (4,5) and (-6,-1)41.58.88.188 (talk) 19:31, 11 July 2017 (UTC)

Find the midpoint between your two points. Draw a line through this midpoint perpendicular to the line between your two points: this line is the locus of points equidistant to your two points. The slope of the line of equidistance is –1 times the reciprocal of slope of the line through your points. Loraof (talk) 19:39, 11 July 2017 (UTC)

To add a bit, first find the slope of the line between the two given points, m (ask if you don't know how). Use the midpoint and the negative reciprocal of the slope, which, as described above, is -1/m, to create the equation of the perpendicular/normal line, in y = -(1/m)x + b form. Once you have this equation of a line, you can plug in any x to find y. If you show your work, we will check it for you. StuRat (talk) 19:57, 11 July 2017 (UTC)

Then midpoint between (4,5) and (-6,-1) is ((4,5) + (-6,-1))/2 = (-1,2). The vector from the midpoint to either one is ±((4,5) - (-6,-1))/2 = ±(5,3). Any (x,y) equidistant to (4,5) and (-6,-1) satisfies ((x,y) - (-1,2))⋅(5,3)=0. That is 5(x+1)+3(y-2)=0 or 5x+3y=1. Bo Jacoby (talk) 21:25, 11 July 2017 (UTC).

You translate the requirement to a formula (sometimes it's an equation, sometimes a system of equations of inequalities...), then solve it. 'Equidistant' means 'in equal distances', so you have

distance of any point (x,y) sought from (4,5) = distance of that (x,y) from (–6,–1)

And this is an equation of the locus you seek. :) You can futher simplify it, if you want (or need).
Plug the well-known formula for Euclidean distance in planar Cartesian coordinates

$d(A,B)={\sqrt {(x_{A}-x_{B})^{2}+(y_{A}-y_{B})^{2}}}$

and you'll get

${\sqrt {(x-4)^{2}+(y-5)^{2}}}={\sqrt {(x-(-6))^{2}+(y-(-1))^{2}}}$

Squaring both sides yields

$(x-4)^{2}+(y-5)^{2}=(x+6)^{2}+(y+1)^{2}$

$x^{2}-8x+16+y^{2}-10y+25=x^{2}+12x+36+y^{2}+2y+1$

Subtract x squared and y squared from both sides

$-8x-10y+41=12x+2y+37$

then organize remaining terms and you'll get an equation of locus, which you will recognize as an equation of a line. --CiaPan (talk) 23:23, 11 July 2017 (UTC)