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 "xaxis", the vertical one is the yaxis.
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 ycoordinate of a point can be defined either as its distance along the yaxis, or as its
perpendicular distance from the xaxis.
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) Coordinate 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 coordinates of C
or x = 2a.
(ii) Section formula:
The point which divides the join of two distinct points A (x_{1}, y_{1})
and B (x_{2}, y_{2}) in the ratio m_{1} : m_{2} Internally, has the coordinates
In particular, the midpoint of the segment joining A (x_{1}, y_{1}) and B (x_{2}, y_{2}) has the coordinates
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 coordinate of the centroid =
y coordinate 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 rightangled triangle.
d(AC)^{2} = d(AB)^{2} + d(BC)^{2}
(5)2 = (3)^{2} + (4)^{2}
Therefore, the triangle is rightangled.
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