Coordinate Geometry - Introduction and Examples (with Solutions) | UPSC Prelims Paper 2 CSAT - Quant, Verbal & Decision Making PDF Download

The X – Y Plane

The number lines, when drawn as shown in X – Y plane below, are called "axes". The horizontal number line
is called the "x-axis", the vertical one is the y-axis.

Abscissa is the x–coordinate of a point can be defined either as its distance along the x–axis, or as its
perpendicular distance from the y–axis.

Ordinate is the y-coordinate of a point can be defined either as its distance along the y-axis, or as its
perpendicular distance from the x-axis.

In the figure given below, OX and OY are two straight lines which are perpendicular to each other and which
intersect at the point O. OX is known as the x–axis, and OY is known as the y–axis. You can see that the two
axes divide the plane into four regions as above. The four regions are known as Quadrants and are named I
Quadrant, II Quadrant, III Quadrant and IV Quadrant as shown.

(A) Co-ordinate of the origin is (0, 0).
(B) Any point on the x axis can be taken as (a, 0)
(C) Any point on the y axis can be taken as (0, b)

Ex.1 In which quadrant is (x, y), such that x y < 0?

Sol. The points (x, y), with xy < 0 means the product of abscissa and ordinate should be negative which can
be possible only when one is positive and other is negative, such as (– 2, 5) or (4, – 6) and this will lie
in the Quadrants II and IV.

(i) Distance formula:


Ex.2 A (a, 0) and B (3a, 0) are the vertices of an equilateral triangle ABC. What are the coordinates of
 C?
 (1) (a, a√3) 

(2) (a√3, 2a) 

(3) (a√3, 0) 

(4) (2a, + a√3) 

(5) None of these
 

Sol. 
Now, the vertex C will be such that AC = BC = 2a
∴ if (x, y) are the co-ordinates of C

or x = 2a.

(ii) Section formula:
The point which divides the join of two distinct points A (x₁, y₁)
and B (x₂, y₂) in the ratio m₁ : m₂ Internally, has the co-ordinates

In particular, the mid-point of the segment joining A (x₁, y₁) and B (x₂, y₂) has the co-ordinates

Ex.3 Find the points A and B which divide the join of points (1, 3) and (2, 7) in ratio 3 : 4 both
 internally & externally respectively .

Sol.

(iii) Centroid and Incentre formulae:
 

Centroid: It is the point of intersection of the medians of a triangle.
 

Incentre: It is the point of intersection of the internal angle bisectors of the angles of a triangle.

If A  be the vertices of a triangle, then its centroid is given by
  and the incentre by, 

Where a = | BC |, b = | CA | and c = | AB |.

Ex.4 If (2, 3), (3, a), (b, – 2) are the vertices of the triangle whose centroid is (0, 0), then find the
 value of a and b respectively.
 (1) – 1, – 4 

(2) – 2, – 5 

(3) – 1, – 6 

(4) – 1, – 5 

(5) None of these

Sol. x co-ordinate of the centroid =

y co-ordinate of the centroid =

a = – 1             Answer: (4)

(iv) Area of triangle:

Ex.5 A triangle has vertices A (2, 2), B (5, 2) and C (5, 6). What type of triangle it is ?

Sol. By the distance formula

According to Pythagorean if the sum of the square of two sides are equal to the square of the third side
then the triangle is a right-angled triangle.
d(AC)² = d(AB)² + d(BC)²
(5)2 = (3)² + (4)²
Therefore, the triangle is right-angled.


Important