My friend asks: find a number twice as far from 15 as it is from 0. (actually, he was using fractions, but let’s find our entertainment elsewhere today).

5, right? But my friend slyly reminds me, and the other answer?

What? -15 is 15 from 0, and -15 is 30 from 15. We have two solutions… In one dimension. And in two dimensions?

What is the locus of all points in the plane twice as far from one given point as from a second given point?

Wait, before you put pencil to paper, or finger tip to mouseclicker, do you agree we will see some sort of conic section? Which do you expect?

(This is totally in the spirit of my bonus question from last Spring)

October 3, 2007 pm31 9:11 pm 9:11 pm

The intuition, as Jonathan says, was some kind of conic. The kind of conic was obvious once written down, and was a surprise to me.

An offshoot problem: Given two points A and B in the co-ordinate plane, find the locus of a third point C such that sin(angle(CAB))=2*sin(angle(CBA)).

Clueless.

2. October 3, 2007 pm31 9:50 pm 9:50 pm

I was surprised too. It was obvious to me that it was going to be symmetric about the X axis, so that left circle or ellipse, but I guessed wrong. It wasn’t hard to work out algebraically, but I admit that my geometry was rusty enough that I had to consult wikipedia to make sure my equation represented what I thought it represented, and figure out how to put it into a standard form. ;-)

Interestingly, an algebraic slip led me to think about the fact that the locus is quite different if you change the question from “twice as far” to “equidistant” ;-)

October 4, 2007 am31 12:57 am 12:57 am

Cute…

4. October 4, 2007 am31 3:49 am 3:49 am

On first thought, I wasn’t expecting an ellipse, even though my mind harkened back to a vague remembrance of eccentricity making me think that the answer had to be either an ellipse or hyperbola.

5. October 4, 2007 am31 7:19 am 7:19 am

Yeah, I forgot that it could have been a hyperbola, and still symmetric about the X-axis.

But I ended up with a circle (radius 10, center (-5,0))

The fact that it needed to remain a different distance from 2 different points made me think ellipse at first. Which wasn’t exactly wrong — a circle is still an ellipse. ;-)