Question

Analytic Geometry PLS HELP!!!
Find an equation for the set of all points whose distance from the origin is twice its distance from x+y-1=0

Answer 1

Let (u,v) be such a point then
\[
\sqrt{u^2+v^2} = 2\frac {|u+v-1|}
{\sqrt{1^2 +1^2}}=2 \frac {|u+v-1|}
{\sqrt{2}}

\]

Answer 2

See http://en.wikipedia.org/wiki/Distance_from_a_point_to_a_line

Answer 3

Can you finish it now?

Answer 4

Square both sides and get
\[
u^2 + v^2 = 2 (u+v-1)^2
\]

Answer 5

\[
u^2+4 u v-4 u+v^2-4 v+2=0
\]

Answer 6

if I use variables x and y:
is x^2=y^2-4x-4y-4=0 the answer?

Answer 7

should be x^2+y^2-4x-4y-4=0

Answer 8

\[
2 - 4 x + x^2 - 4 y + 4 x y + y^2=0
\]

Answer 9

It is a hyperbola. I will post the graph soon.

Answer 10

I just can't understand the thing about the origin... cause we're given a formula... and it goes like this:
(Ax+By+C) / +or - sqrt (A^2+B^2)

Answer 11

Here you can see the line and the graph of the hyperbola. You can see the distance from a point on the hyperbola is twice the distance to the line.

Answer 12

What is the distance from (u,v} to the origin (0,0)?

Answer 13

\[
(u-0)^2 + (v-0)^2=u^2 + v^2
\]
This is the distance formula between two points.

Answer 14

The other is the distance from (u,v) to a line a x + b y + c =0
\[
d=\frac{| au + bv + c|}
{\sqrt{a^2 + b^2}}
\]

Answer 15

I tried to solve it and I got x^2+y^@+4xy-4y-4x+2=0 is it the right answer?

Answer 16

y^2*

Answer 17

yeah and that's what I got after combining similar terms

Answer 18

This is the right answer
\[2 - 4 x + x^2 - 4 y + 4 x y + y^2=0
\]

Answer 19

It seems that you got it right

Answer 20

yeah it's the same thing...I just arranged it

Answer 21

woaAh thanks for your help ^^

Answer 22

Did you see the nice graph, I drew it for you?

Answer 23

I just saw it... you need not to draw it... but thanks for the great effort ^^

Answer 24

Just another confirmation that our work was right