How do you write the equation of the locus of all points in the coordinate plane 8 units from (4, 10)?

1 Answer
KillerBunny
Jun 15, 2016

#(x-4)^2+(y-10)^2=64#, or #x^2+y^2-8x-20y+52=0#

Explanation:

By definition, the locus of all the points who have the same distance from a fixed centre is a circle. So, that was a complex way to ask for a circle of center #(4,10)# and radius #8#.

If you know the center coordinates #(x_0,y_0)# and the radius #r#, the formula for the cirle is given by

#(x-x_0)^2+(y-y_0)^2=r^2#

In your case, #x_0=4#, #y_0=10# and #r=8#, so the expression becomes

#(x-4)^2+(y-10)^2=64#

You can either leave it as it is, or expand the squares and sum all the coefficients:

#x^2-8x+16+y^2-20y+100=64#

and thus

#x^2+y^2-8x-20y+52=0#

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