Co-ordinate Geometry

Introduction

The co-ordinate geometry study material consists of all the important points, definitions of axes, quadrants, sign notations, etc., that are required to measure and locate a particular object. It represents multiple views of an object based on the axes and quadrants selected. In this way, a proper two-dimensional representation of an object is created that provides the basic architectural view for engineering graphics and other designs.

The father of co-ordinate geometry has provided various notations, sign conventions, and reference points that should be used accordingly. Co-ordinate geometry provides an architectural view and implementation details of an object under study. In this way, it provides detailed information and suitable mechanisms for construction and designing.

General Terms

The study material notes on co-ordinate geometry provide details about the basic terminologies used in the co-ordinate geometry. The co-ordinate geometry deals with the 2-D and 3-D representation of objects in the co-ordinate plane. It has various applications that use the distance and area covered by these objects. There are various terms provided by the father of co-ordinate geometry which includes

• Coordinates: The coordinates of an object or diagram are the points that represent that figure. These points have two values referring to the position of the object on the coordinate plane. The coordinates of any point or figure have two values, one referring to the x-axis and the other to the y-axis.
• Coordinate Plane: The coordinate plane is the actual space where the coordinates are plotted. It is made by the intersection of two perpendicular lines, x and y, which are known as the x-axis and y-axis.
• Sign Convention: The intersection of two axes, i.e., x and y, is known as the origin of the coordinate plane. The upper half of the y-axis and the right half of the x-axis represent the positive half of the plane. The lower half of the y-axis and the left half of the x-axis represent the negative half of the plane.
• Quadrant: Based on the sign conventions, the plane has 4 quadrants having different sign combinations for the x-axis and y-axis. For,
  • The upper right half (x,y) = (+,+)
  • The upper left half (x,y) = (-,+)
  • The lower left half (x,y) = (-,-)
  • The lower right half (x,y) = (+,-)

Representation Using the Co-ordinate Geometry

The terms proposed by the father of co-ordinate geometry have provided the basic way to represent a figure on the coordinate plane in the form of straight lines. The curved part can also be represented on the coordinate plane, but it won’t be much effective as compared to the straight lines. The basic idea of using co-ordinate geometry is to use straight lines for various statistical calculations related to the figure. This co-ordinate geometry study material covers all the basic ways of representing a figure that includes

• Representation of a straight line: Kx + Jy + L = 0, where K and J are the numerical values associated with the x-axis and y-axis, respectively, to represent the coordinate values, and L is a constant that can be 0.
• Slope: It provides the inclination value of the figures and is represented by y – Mx = L, where M is a numeric value associated with the x-axis and L is the constant.
• These numeric values associated with the axes are known as intercepts, e.g., in the above two cases, K and M are x-intercepts, and J is the y-intercept.
• To find the value of a particular axis, put the value of the other intercept equal to 0.
  • Take a straight line equation as Qx + Ry + S = 0
  • To get the value of x, put R = 0, therefore, Qx + S = 0, which gives, x = -S/Q
  • Similarly, to get the value of y, put Q = 0, therefore, Ry + S = 0, which gives, y = -S/R

Formulae of Co-ordinate Geometry

The co-ordinate geometry is more like statistical mathematics that needs proper calculations and sign conventions for precise output. It deals with various real-life applications that include different business and statistical domains. For this, there are various theorems and formulas provided by the father of co-ordinate geometry so that various common problems can be solved efficiently. This includes

• The Distance Formula: If two points or coordinates of two lines are given, the distance between them can be found by using this general formula. If the two points have coordinates  (x1,y1) and (x2,y2), respectively, then the distance D is given by
  • D = =√(x2−x1)2+(y2−y1)2
• The Section Formula: If a point Q divides a line and the ratio of internal or external division is known, the coordinates of the point can be found. If the endpoints of the line has coordinates (x1,y1) and (x2,y2), respectively, the ratio of division is a:b, and coordinates of Q are a and y
  • For internal division: (x,y) = ((X2*a + X1*b)/(a+b), (Y2*a + Y1*b)/(a+b))
  • For external division: (x,y) = ((X2*a – X1*b)/(a-b), (Y2*a – Y1*b)/(a-b))

The Mid-Point Theorem: Co-ordinate Geometry

To find the midpoint of a line whose endpoints have coordinates (X1, Y1) and (X2, Y2), respectively, the midpoint theorem can be used. If the coordinates of the midpoint are represented by (x, y), its value is given by;

(x, y) = ((X1 + X2)/2, (Y1 + Y2)/2)

Conclusion

The study material notes on co-ordinate geometry provide details about the formulae and theorem required to calculate the distance and area measures of a particular figure, and the different ways to evaluate the coordinates of required points. This helps in the estimation of various values of a figure, including its location, inclination, reference position, and much more. The basic idea provided by the father of co-ordinate geometry is to represent 3-D and 2-D objects in a plane so that their structure can be evaluated or constructed according to the specified requirements. Co-ordinate geometry is a vast topic that includes various branches of mathematics, majorly statistics, for better evaluation.