# Determining the equation of a curve.

## Homework Statement

A curve is traced by a point P(x,y) which moves such that its distance from the point A(-1,1) is three times its distance from the point B(2,-1). Determine the equation of the curve.

None

## The Attempt at a Solution

Distance from Point A to Point P:
$$\sqrt{(x+1)^{2} + (y-1)^{2}}$$
Distance from Point P to Point B:
$$\sqrt{(x-2)^{2} + (y+1)^{2}}$$

Distance from Point A to Point P is three times the distance from Point P to Point B so...
$$3 \sqrt{(x+1)^{2} + (y-1)^{2}} = \sqrt{(x-2)^{2} + (y+1)^{2}} \\ 9(x+1)^{2} + 9(y-1)^{2} = (x-2)^{2} + (y+1)^{2} \\ 9x^{2} + 18x + 9 + 9y^{2} - 18y - 9 = x^{2} - 4x + 4 + y^{2} + 2y + 2 \\ 8x^{2} + 22x + 8y^{2} - 20y + 13 = 0$$

Doing this gives me the equation of a circle, which I don't think is a curve. After figuring out that the center of that circle was (-11/8, 5/4), the distance from the center to B is not 3 times the distance from the center to A. Then, I think my answer is wrong.

Reanalyzing the problem, I thought of a different approach which was to solve for the equation of a parabola knowing the directrix would be a line going through A(-1,1) and then the focus being (2,-1). However, this would make a slanted parabola and I have no idea how to make an equation for that.

SteamKing
Staff Emeritus
Homework Helper
Arcs and circles are curves. A curve describes any figure which is not a straight line.

Ray Vickson
Homework Helper
Dearly Missed

## Homework Statement

A curve is traced by a point P(x,y) which moves such that its distance from the point A(-1,1) is three times its distance from the point B(2,-1). Determine the equation of the curve.

None

## The Attempt at a Solution

Distance from Point A to Point P:
$$\sqrt{(x+1)^{2} + (y-1)^{2}}$$
Distance from Point P to Point B:
$$\sqrt{(x-2)^{2} + (y+1)^{2}}$$

Distance from Point A to Point P is three times the distance from Point P to Point B so...
$$3 \sqrt{(x+1)^{2} + (y-1)^{2}} = \sqrt{(x-2)^{2} + (y+1)^{2}} \\ 9(x+1)^{2} + 9(y-1)^{2} = (x-2)^{2} + (y+1)^{2} \\ 9x^{2} + 18x + 9 + 9y^{2} - 18y - 9 = x^{2} - 4x + 4 + y^{2} + 2y + 2 \\ 8x^{2} + 22x + 8y^{2} - 20y + 13 = 0$$

Doing this gives me the equation of a circle, which I don't think is a curve. After figuring out that the center of that circle was (-11/8, 5/4), the distance from the center to B is not 3 times the distance from the center to A. Then, I think my answer is wrong.

Reanalyzing the problem, I thought of a different approach which was to solve for the equation of a parabola knowing the directrix would be a line going through A(-1,1) and then the focus being (2,-1). However, this would make a slanted parabola and I have no idea how to make an equation for that.

You have written 3*d(A,P) = d(P,B), the exact opposite of what you want.

• 1 person
$$3 \sqrt{(x+1)^{2} + (y-1)^{2}} = \sqrt{(x-2)^{2} + (y+1)^{2}}$$
You have placed the multiple 3 on the wrong side of the equation here. Try again from that point.

Also, be careful when you expand the expressions as you've made a couple of sloppy errors in your subsequent lines of working too.

EDIT: Beaten to it by Ray!

• 1 person
Wow, I feel stupid for doing that. Finally got it now. Thank you all!