Question

What is the equation of the locus of points at a distance of `sqrt(20)` units from `(0,1)`? What are the coordinates of the points on the line `y=1/2x+1` at a distance of `sqrt(20)` from `(0, 1)`?

Answer 1

Equation: `x^2+(y-1)^2=20`

Coordinates of specified points: `(4,3)` and `(-4,-1)`

Explanation:

Part 1
The locus of points at a distance of `sqrt(20)` from `(0,1)`
is the circumference of a circle with radius `sqrt(20)` and center at `(x_c,y_c)=(0,1)`

The general form for a circle with radius `color(green)(r)` and center `(color(red)(x_c),color(blue)(y_c))` is
`color(white)("XXX")(x-color(red)(x_c))^2+(y-color(blue)(y_c))^2=color(green)(r)^2`

In this case
`color(white)("XXX")x^2+(y-1)^2=20`
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Part 2
The coordinates of the points on the line `y=1/2x+1` at a distance of `sqrt(20)` from `(0,1)`
are the intersection points of
`color(white)("XXX")y=1/2x+1` and
`color(white)("XXX")x^2+(y-1)^2=20`

Substituting `1/2x+1` for `y` in `x^2+(y-1)^2=20`
`color(white)("XXX")x^2+(1/2x)^2=20`

`color(white)("XXX")5/4x^2=20`

`color(white)("XXX")x^2=16`

Either
`color(white)("XXX")x=+4color(white)("XXX")rarry=1/2(4)+1=3`
or
`color(white)("XXX")x=-4color(white)("XXX")rarry=1/2(-4)+1=-1`