Chapter 14 B: Coordinate Geometry

• Let X'OX and Y’OY be two mutually perpendicular lines through any point O in the plane of the paper. Point O is known as the origin. The line X'OX is called the x-axis or axis of x; the line Y'OY is known as the y-axis or axis of y; and the two lines taken together are called the coordinate axes or the axes of coordinates.





• Any point can be represented on the plane described by the coordinate axes by specifying its x and y coordinates. The x coordinate of the point is also known as the abscissa while the y coordinate is also known as the ordinate.




• Distance Formula : If two points P and Q are such that they are represented by the points (x₁, y₁) and (x₂, y₂) on the x-y plane (cartesian plane), then the distance between the points P and Q = \( \sqrt{(x_1 - x_2)^2 - (y_1 - y_2)^2} \)




• Section Formula : If any point (x, y) divides the line segment joining the points (x₁, y₁) and (x₂, y₂) in the ratio m : n internally, then x = (mx₂ + nx₁)/(m + n) and y = (my₂ + ny₁)/(m - n)





• If any point (x, y) divides the line segment joining the points (x₁, y₁) and (x₂, y₂) in the ratio m : n externally, then x = (mx₂ - nx₁)/(m + n) and y = (my₂ - ny₁)/(m - n)

Area of a Triangle

• The area of a triangle whose vertices are A (x₁, y₁), B (x₂, y₂) and C (x₃, y₃) is given by




• \( \frac{x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)}{2} \)




• Since the area cannot be negative, we have to take the modulus value given by the above equation




• If one of the vertices of the triangle is at the origin and the other two vertices are A (x₁, y₁), B (x₂, y₂) , then the area of triangle is \( \frac{(x_1y_2 - x_2y_1)}{2} \)

Centre of gravity or centroid of a triangle

• The centroid of a triangle is the point of intersection of its medians (the line joining the vertex to the middle point of the opposite side).




• Centroid divides the medians in the ratio 2 : 1.




• In other words, the CG or the centroid can be viewed as a point at which the whole weight of the triangle is concentrated




• If A (x₁, y₁), B (x₂, y₂) and C (x₃, y₃) are the coordinates of the vertices of a triangle, then the coordinates of the centroid G of that triangle are x = (x1 + x2 + x3)/3 and y = (y1 + y2 + y3)/3

In-centre of a triangle

• The centre of the circle that touches the sides of a triangle is called its Incentre. In other words, if the three sides of the triangle are tangential to the circle then the centre of that circle represents the in-centre of the triangle




• The in-centre is also the point of intersection of the internal bisectors of the angles of the triangle. The distance of the in-centre from the sides of the triangle is the same and this distance is called the in-radius of the triangle.




• IfA (x₁, y₁), B (x₂, y₂) and C (x₃, y₃) are the coordinates of the vertices of a triangle, then the coordinates of its in-centre are




• \( x = \frac{ax_1 + bx_2 + cx_3}{a+b+c}, y = \frac{ay_1 + by_2 + cy_3}{a+b+c} \) where BC = a, AB = c and AC = b.

Circumcentre of a triangle

• The point of intersection of the perpendicular bisectors of the sides of a triangle is called its circumcentre




• It is equidistant from the vertices of the triangle.




• It is also known as the centre of the circle which passes through the three vertices of a triangle (or the centre of the circle that circumscribes the triangle.)




• Let ABC be a triangle. If O is the circumcentre of the triangle ABC, then OA = OB = OC and each of these three represent the circum radius.

Orthocentre of a triangle

• The orthocentre of a triangle is the point of intersection of the perpendiculars drawn from the vertices to the opposite sides of the triangle.

Collinearity of three points

• Three given points A, B and C are said to be collinear, that is, lie on the same straight line, if any of the following conditions occur:




• Area of triangle formed by these three points is zero




• Slope of AB = Slope of AC




• Any one of the three points (say C) lies on the straight line joining the other two points (here A and B).

Slope of a line

• The slope of a line joining two points A (x₁, y₁), B (x₂, y₂) is denoted by m and is given by m = (y₂ – y₁)/(x₂ – x₁) = tan q, where q is the angle that the line makes with the positive direction of x-axis.




• This angle q is taken positive when it is measured in the anti-clockwise direction from the positive direction of the axis of x

• General Form : The general form of the equation of a straight line is ax + by + c = 0. (First degree equation in x and y). Where a, b and c are real constants and a, b are not simultaneously equal to zero. In this equation, slope of the line is given by -a/b




• The general form is also given by y = mx + c; where m is the slope and c is the intercept on y-axis. In this equation, slope of the line is given by m.




• Line Parallel to the X-axis : The equation of a straight line parallel to the x-axis and at a distance b from it, is given by y = b. Obviously, the equation of the x-axis is y = 0




• Line Parallel to Y-axis : The equation of a straight line parallel to the y-axis and at a distance a from it, is given by x = a. Obviously, the equation of y-axis is x = 0




• Slope Intercept Form : The equation of a straight line passing through the point A (x₁, y₁) and having a slope m is given by (y – y₁) = m (x – x₁)




• Two Points Form : The equation of a straight line passing through two points A (x₁, y₁), B (x₂, y₂) is given by \( (y - y_1) = \frac{(y_2 - y_1)(x - x_1)}{x_2 - x_1} \). Its slope = \( \frac{y_2 - y_1}{x_2 - x_1} \)




• Intercept Form The equation of a straight line making intercepts a and b on the axes of x and y respectively is given by x/a + y/b = 1. If a straight line cuts x-axis at A and the y-axis at B then OA and OB are known as the intercepts of the line on x-axis and y-axis respectively

• Condition for two lines to be parallel: Two lines are said to be parallel if their slopes are equal.




• For this to happen, ratio of coefficient of x and y in both the lines should be equal.




• In a general form, this can be stated as: line parallel to ax + by + c = 0 is ax + by + k = 0 or dx + ey + k = 0 if a/d = b/e where k is a constant




• Length of perpendicular or Distance of a point from a line : The length of perpendicular from a given point (x₁, y₁) to a line ax + by + c = 0 is \( \frac{ax_1 + by_1 + c}{\sqrt{a^2 + b^2}} \)

Distance between two parallel lines.

• If two lines are parallel, the distance between them will always be the same.




• When two straight lines are parallel whose equations are ax + by + c = 0 and ax + by + c1 = 0, then the distance between them is given by




• \( \frac{c - c_1}{\sqrt{a^2 + b^2}} \)




• The length of the perpendicular from the origin to the line ax + by + c = 0 is given by \( \frac{c}{\sqrt{a^2 + b^2}} \)

Change of axes

• If origin (0, 0) is shifted to (h, k) then the coordinates of the point (x, y) referred to the old axes and (X, Y) referred to the new axes can be related with the relation x = X + h and y = Y + k





• Point of intersection of two line : Point of intersection of two lines can be obtained by solving the equations as simultaneous equations.




• If all the three vertices of a triangle have integral coordinates, then that triangle cannot be an Equilateral triangle.