COORDINATE GEOMETRY
FUNDAMENTALS
- Co-ordinate Geometry-The branch of mathematics in which Geometric problems are solved through algebra by using the co-ordinates system is known as co-ordinate Geometry.
- The horizontal line XOX' is called x-axis and the vertical line YOY' is called the y-axis.
- The point O is called the origin and co-ordinates of-the origin are (0, 0)
- he distance of a point from y-axis is called a x-coordinate or abscissa
- The distance of the point from x axis is called its y-coordinate or ordinate.
- The y co-ordinate of every point on x axis is zero. So, the co-ordinates of any point on the x axis are of the form (x, 0)
- The x co-ordinate of every point on y axis is zero. So, the co-ordinates of any point on y- axis are of the form (0,y)
- \[x=0\], denotes y axis
- \[y=0\], denotes x axis
- \[y=a\], where a is constant denotes a straight line Parallel to x axis
- \[x=a\], where a is constant denotes a straight line parallel to y axis
Quadrant:-
|
Quadrant
|
Nature of a and y
|
Sign of Co-ordinates
|
|
I
|
x>0, y>0
|
\[(+,+)\]
|
|
II
|
x<0, y>0
|
\[(-,+)\]
|
|
III
|
x<0, y<0
|
\[(-,-)\]
|
|
IV
|
x>0, y<0
|
\[(+,-)\]
|
- Distance formula: Distance between two points \[\left( {{x}_{1}},{{y}_{1}} \right)\] and \[({{x}_{2}},{{y}_{2}})\]\[=\sqrt{{{\left( {{x}_{2}}-{{x}_{1}} \right)}^{2}}+{{\left( {{y}_{2}}-{{y}_{1}} \right)}^{2}}}\]
- Mid-point of two points \[\left( {{x}_{1}},{{y}_{1}} \right)\] and \[({{x}_{2}},{{y}_{2}})\]\[\left( \frac{{{x}_{1}}+{{x}_{2}}}{2},\frac{{{y}_{1}}+{{y}_{2}}}{2} \right)\]
- Area of \[\Delta \] with coordinates \[\left( {{x}_{1}},{{y}_{1}} \right)\], \[({{x}_{2}},{{y}_{2}})\] and \[({{x}_{2}},{{y}_{3}})\]
\[\frac{1}{2}\left[ {{x}_{1}},\left( {{y}_{2}}-{{y}_{3}} \right)+{{x}_{2}}\left( {{y}_{3}}-{{y}_{1}} \right)+{{x}_{3}}\left( {{y}_{3}}-{{y}_{1}} \right) \right]\]
- Centroid of a triangle is the point of intersection of its medians.
- Centroid of a triangle of vertices \[\left( {{x}_{1}},{{y}_{1}} \right)({{x}_{2}},{{y}_{2}})\] and \[({{x}_{3}},{{y}_{3}})\] is \[\left( \frac{{{x}_{1}}+{{x}_{2}}+{{x}_{3}}}{3},\frac{{{y}_{1}}+{{y}_{2}}+{{y}_{3}}}{3} \right)\
