Introduction to Co-ordinate Geometry

Introduction to co-ordinate geometry:
Co-ordinate geometry is a branch of Mathematics that studies about points, lines and geometrical figures using co-ordinate systems. In geometry, we study the same using geometrical constructions and actual measurement but in co-ordinate geometry it is predominantly using co-ordinates of points.

Some of the topics covered in co-ordinate geometry are  finding distance between two points, slope of line, equations of lines, circles and geometrical figures etc

Let us solve some of co-ordinate geometry problems to get a feel of the subject

Understanding Vertex Form of Parabola is always challenging for me but thanks to all math help websites to help me out.

Solve Points Problems of Co-ordinate geometry:


Problem 1:

Find the slope of the line for these two points. ( 5 , 6 ) and ( -3 , 9 )

Solution:

The slope formula is given by

m = `(y2-y1)/(x2-x1)`

Given two points: (x1, y1) =  (5, 6) and (x2, y2) = (-3, 9)

Apply these two points into that formula for finding slope.

m = `(9-6)/(-3-5)`

m = `3/-8`

m = `- 3/8`

The slope of these two points is `-3/8` .

Problem 2:

Solve the distance for the given two points (5, 4) and (4,6)

Solution:

The distance formula is given by

d = `sqrt((x2-x1)^2+(y2-y1)^2)`

d = `sqrt((4-5)^2+(6-4)^2)`

d = `sqrt((-1)^2+(2)^2)`

d = `sqrt(1+ 4)`

d = `sqrt(5)`

The distance for these points is 2.23.


Solve Lines problem of Co-ordinate geometry:


Problem:

Solve the equation of a line between these two points (3, 8) and (-6, 4).

Solution:

Line equation form is y = mx + b

Solve m, slope between these two points.

The slope formula is given by

m = `(y2-y1)/(x2-x1)`

Given two points: (x1, y1) =  (3, 8) and (x2, y2) = (-6, 4)

Apply these two points into that formula for finding slope.

m = `(4-8)/(-6-8)`

m = `-4/-14`

m = `4/14` ---------------Simplify it.

m = `2/7`

Solve b, y intercept

For one point (x1, y1) = (3,8), the line equation becomes

y1 = mx1 + b

8  = `2/7` (3) + b

8  =  `6/7`   + b

b = 8 – `6/7`

b = `50/7`

Substitute m and b into line equation, we get

y = mx + b

y = `2/7` x + `50/7`

y = `1/7` (2x+50)

Multiply by 7 both on sides,

7y = 2x + 50

2x – 7y +50 = 0.

So the equation of line between these two points is 2x – 7y + 50 = 0.


Solve Circle Problem of Co-ordinate geometry:


Problem:

Find the center and radius of (x – 5)^2 + (y – 7)^2 = 25 circle.

Solution:

The circle equation form is (x-h)^2 + (y-k)^2 = R2.

Here the center is (h, k) and Radius is R.

The given equation looks the same as circle equation form.

(x-5)^2 + (y-7)^2 = 52

From the given equation, the center is (5, 7) and Radius is 5.