What is the equation of the locus of points at a distance of sqrt(20)20 units from (0,1)(0,1)? What are the coordinates of the points on the line y=1/2x+1y=12x+1 at a distance of sqrt(20)20 from (0, 1)(0,1)?

1 Answer
Jun 2, 2016

Equation: x^2+(y-1)^2=20x2+(y1)2=20

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

Explanation:

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

The general form for a circle with radius color(green)(r)r and center (color(red)(x_c),color(blue)(y_c))(xc,yc) is
color(white)("XXX")(x-color(red)(x_c))^2+(y-color(blue)(y_c))^2=color(green)(r)^2XXX(xxc)2+(yyc)2=r2

In this case
color(white)("XXX")x^2+(y-1)^2=20XXXx2+(y1)2=20
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Part 2
The coordinates of the points on the line y=1/2x+1y=12x+1 at a distance of sqrt(20)20 from (0,1)(0,1)
are the intersection points of
color(white)("XXX")y=1/2x+1XXXy=12x+1 and
color(white)("XXX")x^2+(y-1)^2=20XXXx2+(y1)2=20

Substituting 1/2x+112x+1 for yy in x^2+(y-1)^2=20x2+(y1)2=20
color(white)("XXX")x^2+(1/2x)^2=20XXXx2+(12x)2=20

color(white)("XXX")5/4x^2=20XXX54x2=20

color(white)("XXX")x^2=16XXXx2=16

Either
color(white)("XXX")x=+4color(white)("XXX")rarry=1/2(4)+1=3XXXx=+4XXXy=12(4)+1=3
or
color(white)("XXX")x=-4color(white)("XXX")rarry=1/2(-4)+1=-1XXXx=4XXXy=12(4)+1=1