Coordinate Geometry
1. Distance formula
If two points P(x1, y1), Q(x2, y2) then the distance between the points P and Q,
D=sqrt [(x1-x2)^2 +(y1-y2)^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,
X = (mx2+nx1) / (m + n)
Y = (my2+ny1) / (m + n)
If externally,
X = (mx2-nx1) / (m-n)
Y = (my2-ny1) / (m-n)
3. Area of Triangle
The area of triangle whose vertices are A(x1, y1), B(x2,y2) & C(x3,y3) is given by
[x1 (y2-y3) +x2 (y3-y1) +x3 (y1-y2)] /2
Note: Since the area cannot be negative, we have to take the modulus value given by the above equation.
a) If one of the vertices of triangle is at the origin and the two vertices are A(x1, y1), B(x2, y2)
Area= [x1y2-x2y1]/2
4. Centroid of a Triangle
It is the point of intersection of its medians. Centroid divides the medians in the ratio 2:1.
If A(x1, y1), B(x2, y2) & 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 & y= (y1+y2+y3)/3
5. In-centre of a Triangle
It is the centre of a circle that touches the side of a triangle is called its In-centre. 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.
If A(x1, y1), B(x2,y2) & C(x3,y3) are the coordinates of the vertices of a triangle then the coordinates of its in-centre are
X= (ax1+bx2+cx3)/ (a +b +c) Y= (ay1+by2+cy3)/ (a +b +c)
Where BC=a, AB=c & AC=b
6. Circum centre of a Triangle
The point of intersection of the perpendicular bisector of the sides of a triangle is called its circum-centre. It is equidistant from the vertices of the triangle. It is also known as the centre of the circle that circumscribes the triangle.
Let ABC be a triangle. If O is the circum-centre of the triangle ABC, then OA=OB=OC and each of these three represent the circum-radius.
7. Collinearity of Three Points
Three given points A, B & C are said to be collinear, that is lie on the same straight line, if any of the following condition occur:
a) Area of triangle formed by these three points is Zero.
b) Slope of AB=slope of AC
8. Slope of a Line
The slope of a line joining two points A(x1, y1) and B(x2, y2) is denoted by m and is given by,
M= (y2-y1)/(x2-x1) = tan z where z is the angle that the line makes with the positive direction of x-axis.
9. Different forms of the equation of a straight line
a) General form
The general form of the equation of a straight line is
ax + by+ c=0
Where a, b and c are real constants.
Slope of line= -a/b
The general form is also given by
Y=mx+ c where m is the slope & c is the intercept on y-axis.
b) 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.
Equation of the x-axis is y=0
c) 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.
Equation of the y-axis is x=0
d) 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)
e) Two points form
The equation of a straight line passing through two points A(x1, y1) and B(x2, y2) is given by
(y- y1)= [(y2-y1) (x-x1)]/(x2-x1)
Slope= (y2-y1)/(x2-x1)
f) Intercept Form
The equation of a straight line making intercepts a and b on the axes of x & y respectively is given by,
x /a + y/b =1
10) Condition for Two lines to be parallel
Two lines are said to be parallel if their slopes are equal. For this to happen, the ratio of coefficients 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.
11) Condition for two lines to be perpendicular
Two lines are said to be perpendicular if product of the slopes of the lines is equal to -1.
12) Length of Perpendicular of a Point from a Line.
The length of perpendicular from a given point (x1,y1) to a line ax +by +c=0 is
|ax1+by1+c|/sqrt (a^2+b^2)
a) Distance b/w two parallel lines will always be the same.
When two straight lines are parallel whose equation are ax +by +c=0 & ax+by+c1=0
Distance b/w them = |c-c1|/sqrt (a^2+b^2)