Coordinate Geometry
Cartesian Coordinate system:
XOX’ and YOY’ are the mutually perpendicular lines on a plane of paper intersecting at O.
The line XOX’ is called the X-axis while the line YOY’ is called the Y-axis. These two axes taken together are called co-ordinate axes and this system is called the Cartesian coordinate system.
Quadrant I:
The points in the first quadrant will be in the form of (x,y) where both x and y are positive.
Quadrant II:
The points in the second quadrant will be in the form of (x,y) where x is negative and y is positive.
Quadrant III:
The points in the third quadrant will be in the form of (x,y) where both x and y are negative.
Quadrant IV:
The points in the fourth quadrant will be in the form of (x,y) where x is positive and y is negative.
(x,y) are called the coordinates of the axis where x-coordinate is called the abscissa and the y-coordinate is called the ordinate.
- Distance between any two points $$(x_1,y_1)$$ and $$(x_2,y_2)$$:
$$AB = |\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}|$$ - Distance between a point (x,y) and origin (0,0) = $$\sqrt{x^2+y^2}$$
- Distance between two polar coordinates $$(r_1,\theta_1)$$ and $$(r_2,\theta_2)$$ = $$\sqrt{r_1^2+r_2^2-2r_1r_2cos(\theta_1-\theta_2)}$$
Let A, B and C be the vertices of a triangle.
AB, BC and CA are the distance between A and B, B and C, C and A respectively.
The triangle is collinear if:
- The sum of any two distances is equal to the third.
AB+BC = AC
AB+AC = BC
BC+AC = AB - The area of the triangle is zero.
- Slope of AB = Slope of BC = Slope of C
Some Important Observations:
- A triangle is called an equilateral triangle if AB = BC = CA
- A triangle is called an isosceles triangle if AB = AC (or) AB = BC (or) AC = BC
- A triangle is called a right-angled triangle if $$AB^2 + BC^2 = AC^2$$
- The coordinates of a point P(x,y) dividing a line segment joining the points $$A(x_1,y_1)$$ and $$B(x_2,y_2)$$ internally in the ratio m:n are:
$$(\dfrac{mx_2+nx_1}{m+n}, \dfrac{my_2+ny_1}{m+n})$$ - If P is the midpoint in the above formula, then the coordinates of P are:
$$(\dfrac{x_1+x_2}{2},\dfrac{y_1+y_2}{2})$$ - The coordinates of a point P(x,y) dividing a line segment joining the points $$A(x_1,y_1)$$ and $$B(x_2,y_2)$$ externally in the ratio m:n are:
$$(\dfrac{mx_2-nx_1}{m-n}, \dfrac{my_2-ny_1}{m-n})$$ - x-axis divides the line joining $$P(x_1,y_1)$$ and $$Q(x_2,y_2)$$ in the ratio $$-y_1:y_2$$.
- y-axis divides the line joining $$P(x_1,y_1)$$ and $$Q(x_2,y_2)$$ in the ratio $$-x_1:x_2$$.
- A line ax+by+c divides another line joining $$P(x_1,y_1)$$ and $$Q(x_2,y_2)$$ in the ratio $$-(ax_1+by_1+c) : (ax_2+by_2+c)$$.
- Area of a triangle whose vertices are $$A(x_1,y_1), B(x_2,y_2), C(x_3,y_3)$$ is:
$$\dfrac{1}{2}\times[x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2)]$$ - Area of a polygon whose vertices are $$(x_1,y_1), (x_2,y_2), (x_3,y_3),...(x_n,y_n)$$ is:
$$\dfrac{1}{2}\times[(x_1y_2-x_2y_1)+(x_2y_3-x_3y_2)+....+(x_ny_1-x_1y_n)]$$ - Centroid of a triangle whose vertices are $$A(x_1,y_1), B(x_2,y_2), C(x_3,y_3)$$ is:
$$(\dfrac{x_1+x_2+x_3}{3},\dfrac{y_1+y_2+y_3}{3})$$.
