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: ( , , )




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.



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)

\sqrt{(h+1)^2 +(k-2)^2} = \sqrt{√(h-4)^2+(k-0)^2}

(h+1)^2 +(k-2)^2 = (h-4)^2+(k-0)^2

h^2+2h+1 + k^2 -4k +4 = h^2 -8h + 16 +k^2

2h+1-4k +4 +8h - 16 = 0

10h -4k - 11 =0

So locus of point p(h,k) is 10x - 4y -11 =0

Some important formulas of coordinate geometry

Some important formulas of coordinate geometry :

Distance between two points (x_1,y_1) and (x_2,y_2)

= \sqrt{(x_2-x_1)^2+(y_2-y_1)^2}


Distance of a point (x_1,y_1) from the center (0,0)

= \sqrt{x_1^2+y_1^2}


Coordinate of a point , who divides a line joining the two points (x_1,y_1) and (x_2,y_2) in ratio of m_1 : m_2

Internal division (\frac{m_1x_2+m_2x_1}{m_1+m_2},\frac{m_1y_2+m_2y_1}{m_1+m_2})

External division (\frac{m_1x_2-m_2x_1}{m_1-m_2},\frac{m_1y_2-m_2y_1}{m_1-m_2})


Mid point formula : Mid point coordinate of a  line segment by joining two points (x_1,y_1) and (x_2,y_2) is :




Distance between Two points (Two Dimensions)

Distance between Two points (Two Dimensions) :

Let P(x_1,y_1) and Q(x_2,y_2) are two points in a plane,distance between them is d.

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

distance formula

According to the above figure:

OM =x_1 ; PM = y_1

ON = x_2 ; QN = y_2

So PR = MN = ON – OM = x_2 -x_1

And QR = QN – RN  = QN – PM = y_2 -y_1

Now In right triangle PRQ :

PQ = \sqrt{PR^2 + QR^2}

Or PQ = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}

Or d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}  ….( 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 Rectangular coordinate axes or rectangular axes.

coordinate axes

Line XOX’ is known as x-axes and line YOY’ is known as y-axes and intersecting point is O, we call it 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 y-coordinate or ordinate of point P. and these x and y are the coordinates of the Point P. we write them like (x,y).


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’.