Question

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

Answer 1

`(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`