A point P:(x,y) moves according to the given law.Find the equation of its locus,?

The distance between a point (x,y) and a given point (a,b) is:

d = √( (x - a)² + (y - b)² )

1.

The distance to the point P₁= (2,-1) is

d₁ = √( (x - 2)² + (y + 1)² )

The distance to the point P₂ = (3,2) is

d₂ = √( (x - 3)² + (y - 2)² )

Equidistant means the two distances are equal:

√( (x - 2)² + (y + 1)² ) = √( (x - 3)² + (y - 2)² )

<=>

(x - 2)² + (y + 1)² = (x - 3)² + (y - 2)²

<=>

x² - 4∙x + 4 + y² + 2∙y + 1 = x² - 6∙x + 9 + y² - 4∙y + 4

<=>

2∙x + 6∙y = 8

The locus is a straight line.

2.

The distance to the point P₁= (4,0) is

d₁ = √( (x - 4)² + (y - 0)² ) = √( (x - 4)² + y² )

The distance to the point P₂ = (1,0) is

d₂ = √( (x - 1)² + (y - 0)² ) = √( (x - 1)² + y² )

The distance to the 1st points is twice the distance to the 2nd point means.

√( (x - 4)² + y² ) = 2∙√( (x - 1)² + y² )

<=>

(x - 4)² + y² = 4∙( (x - 1)² + y² )

<=>

x² - 8∙x + 16 + y² = 4∙(x² - 2∙x + 1 + y²)

<=>

x² - 8∙x + 16 + y² = 4∙x² - 8∙x + 4 + 4∙y²

<=>

3∙x² + 3∙y² = 12

<=>

x² + y² = 4

The locus is a circle centered at the origin with the radius 2

3.

The distance to the point P₁= (0,-4) is

d₁ = √( (x - 0)² + (y + 4)² ) = √( x² + (y + 4)² )

The distance to the point P₂ = (0,-9) is

d₂ = √( (x - 0)² + (y + 9)² ) = √( x² + (y + 9)² )

The distance to the 1st points is two third the distance to the 2nd point means.

√( x² + (y + 4)² ) = (2/3)∙√( x² + (y + 9)² )

<=>

x² + (y + 4)² = (4/9)∙( x² + (y + 9)² )

<=>

9∙(x² + y² + 8∙y + 16) = 4∙(x² + y² + 18∙y + 81)

<=>

9∙x² + 9∙y² + 72∙y + 144 = 4∙x² + 4∙y² + 72∙y + 244

<=>

5∙x² + 5∙y² = 100

<=>

x² + y² = 20

The locus is a circle centered at the origin with the radius √20