## Archive for the ‘Co ordinate Geometry’ Category

## Problem 673

What are the projections of the point (-2,3,10) on the coordinate planes?

On the xy-plane: ( , , )

On the yz-plane: ( , , )

On the xz-plane: ( , , )

**Solution:-**

On the xy-plane(-2,3,0)

On the yz-plane(0,3,10)

On the xz-plane (-2,0,10)

## Locus of a point (mathematics)

**Locus :-**

If a point travels according to the given condition then path created by point is known as the locus of that point.

Or locus is a geometric shape where every point satisfies the given condition.

**Equation of the locus: **

A algebraic relations got from the geometric condition for any variable point (x,y) is known as equation of the locus.

** **

**Example:-**

**Find the locus of a point, which is always at equal distance from two points (-1,2) and (4,0).**

** **

Let the coordinate that point is P(h,k).

Now according to the given condition

Distance from p(h,k) to (-1,2) = distance from P(h,k) to (4,0)

So locus of point p(h,k) is

## Some important formulas of coordinate geometry

**Some important formulas of coordinate geometry :**

Distance between two points and

=

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Distance of a point from the center (0,0)

=

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Coordinate of a point , who divides a line joining the two points and in ratio of

Internal division

External division

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Mid point formula : Mid point coordinate of a line segment by joining two points and is :

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## Distance between Two points (Two Dimensions)

**Distance between Two points (Two Dimensions) :**

Let and are two points in a plane,distance between them is .

Now we have to find the distance , draw two perpendiculars on axes from P and Q points, which are PM and QN, now draw a perpendicular PR from point P on QN.

According to the above figure:

OM = ; PM =

ON = ; QN =

So PR = MN = ON – OM =

And QR = QN – RN = QN – PM =

Now In right triangle PRQ :

PQ =

Or PQ =

**Or d = ….( distance formula)**

## Introduction of co-ordinate Geometry

First it was studied by French Mathematician Rene Descartes in 1637. In this geometry , position of the points is denoted by some special numbers we call them coordinates and we denote different shapes ( lines, circles etc ) with the help of algebraic equations. We use coordinates in this branch of mathematics, so we call it Co-ordinate geometry.

**Cartesian Coordinates:**

We draw two perpendicular lines in a plane, both are intersecting to each other at a point O, horizontal line is XOX’ and vertical line is YOY’, we call them ** Coordinate axes** or

**or**

*Rectangular coordinate axes***.**

*rectangular axes*Line XOX’ is known as ** x-axes** and line YOY’ is known as

**and intersecting point is O, we call it**

*y-axes***.**

*origin*Let there is a point P in a plane, draw two perpendiculars from point P to x and y axes. Which are PM and PN ( shown in the below figure ), we denote the distance of P point from x-axes is x and from y-axes is y. we call x as ** x-coordinate or abscissa** and y as

**of point P. and these x and y are the coordinates of the Point P. we write them like (x,y).**

*y-coordinate or ordinate***Quadrant and sign of coordinates:**

Coordinate axes divides the plane into four parts, these four parts are XOY, YOX’, X’OY’ and Y’OX. We call them first, second, third and fourth quadrant.

From the right and upper side of the origin all distances are positive on OX and on OY and at left and down side from the origin all distances are negative on OX’ and on OY’.