Analytic Geometry in the Plane

 

Consider the Cartesian rectangular coordinate system in the plane. Taking the projection of any point М₁ on the x and y coordinatewe obtain two numbers x=a₁ and y=b₁. Take two numbers a₂ and b₂ plot a₂ on the x -axis and b₂ on the y -axis. Having drawn two straight lines parallel to the coordinate axes through these points we find obtain a point M ₂ in their intersection.

a2

M2(a2;b2)

b2

M1(a1;b1) b1

		x

0

a1

Thus, there is a one–to–one correspondence between points in the plane and pairs of numbers.

The distance between two points. Let us find the distance between two points М ₁ and М ₂ in the plane.

 

y

M₂(x_2;,y₂)

d

M₁(x₁;y₁)

0 x

Compose the vector .

The length of this vector is defined by

 

.

 

This is the distance between the two given points.

Example. Find the distance between the two points А (2;3) and В (–4;11).

Using the above formula, we obtain

.

Division of an interval in a given ratio. Suppose given an interval М ₁ М ₂. Let us find the coordinates a of point М on the interval for which .

Compose the vectors and .

y

M₂(x₂,y₂)

M(x,y)

M₁(x₁,y₁)

0 x

This gives the x coordinate; y is found in a similar way:

; .

To obtain a formula for the midpoint of the interval, we take l =1:

; .

Example. Given the two points М ₁(–2;4) and М ₂(6;2), find the midpoint of the interval М ₁, М ₂.

М ₁ ,

М ₂ .