Question

Find coordinates for a point that is three times as far from the origin as (2, 3) is. Describe the configuration of all such points.

Answer 1

@phi

Answer 2

do you know how far away (2,3) is from the origin ?
(I would use the "distance formula")

Answer 3

\(\sqrt{13}\)

Answer 4

and three times that distance is
\[ 3 \sqrt{13} \]
they want a point that far from the origin
I can think of a few "easy" points. For example, make the y value 0, what x value should we use ?

Answer 5

\((0, 3\sqrt{13})\)

Answer 6

Or \((\pm 3\sqrt{13}, 0)\)

Answer 7

Let me make the \((0, 3\sqrt{13})\) into \((0, \pm 3\sqrt{13})\)

Answer 8

We could make a circle right by connecting all four of those points?

Answer 9

ok, that is on the y-axis, but it works also

and they also want to know what the "locus of points"

i.e. all points equidistant from a common point (also known as the "center")

Answer 10

The radii would all be \(3\sqrt{13}\)

Answer 11

yes, a circle is made up of points equally far from the center (the origin in this case)

Answer 12

So would be have something like \(x^2 + y^2 = (3\sqrt{13})^2\)?

Answer 13

you could, but they don't ask for an equation
(but giving the equation is ok)

Answer 14

Ok. So what do they want when they mean "describe the configuration"?

Answer 15

I think they want:
All points 3 times farther from the origin than the point (2,3) form a circle, with the center at (0,0), and radius 3 sqr(13)

Answer 16

Yes that makes so much sense :) Thank you!