Some important formulae for straight lines from Co-ordinate Geometry :
1. Distance between points A(x1,y1) and B(x2,y2) : AB = rt {(x2-x1)^2+(y2-y1)^2}
2.Area of a triangle ABC formed by three co-ordinates A(x1,y1), B(x2,y2) and C(x3,y3)
ar(ABC) = 1/2[(x1y2-x2y1)+(x2y3-y2x3)+(x3y1-x1y3)]…..
To remember this formula… follow the trick given below:
(i) write x1, x2, x3,x1 in one column facing opp to y1, y2, y3, y1 respectively….
(ii) Match diagonally x1…y2, y1…x2, y2…x3, x2…y3, y3…x1, x3…y1 and take product of
diagonal pairs.
(iii) Take sign negative while moving for product diagonally upwards and positive for downwards.
(iv) Sum the products and divide by 2.
3. Centroid of triangle ABC is
G{ (x1+x2+x3)/3 , (y1+y2+y3)/3 }
4. Incentre of the triangle ABC, if AB = c, AC= b, BC= a, is
{ (ax1+bx2+cx3)/(a+b+c) , (ay1+by2+cy3)/(a+b+c) }
5. (i) If point P(a,b) divides AB internally in ratio AP : PB = m1:m2,
where A(x1,y1) & B(x2,y2) join to form AB, then :
a = (m1x2+m2x1)/(m1+m2) & b = (m1y2+m2y1)/(m1+m2)
(ii) If P divides AB externally in the ratio m1:m2 (i.e. extend AB to P such that AP:BP = m1:m2)
a = (m1x2-m2x1)/(m1-m2) & b = (m1y2-m2y1)/(m1-m2)
6. Slope ‘m’ of a straight line :
It’s the ‘tan’ value of angle @ which the line makes with the positive direction of x-axis.
So if AB be a line joining A(x1,y1) and B(x2,y2) and making angle @ with the +ve direction of x-axis
Then, m = tan @ = (y2-y1)/(x2-x1)…. Where x1 is not equal to x2
Following results are important here :
(i) If x1=x2, then line is perpendicular to x-axis. ……. Because in that case tan@= infinity
(ii) Slope of line parallel to x-axis is zero. ………………here tan@=0
(iii) If the line makes acute angle with x-axis its slope is positive…….. tan@ is +ve for acute angle.
(iv) If the line makes obtuse angle with the x-axis, its slope is negative. …… tan@ is negative for
obtuse angle.
(v) when two lines are parallel their slopes m1 & m2 are equal……. m1=m2
(vi) when two lines are perpendicular to each other m1*m2 = -1
7. Equations of Line : ( Any expression of the form Ax+By+C=0 represent the eqn of straight line
Provided both A & B are not equal to zero)
(i) Two Point form : Eqn of a line joining two points (x1,y1) and (x2,y2) is given by :
(y-y1) = { (y2-y1)/(x2-x1)} (x-x1)
(ii) Point Slope form : If a line is of slope ‘m’ and passes through (x1,y1), the eqn of line is :
(y-y1) = m(x-x1)
(iii) Intercept form : If a straight line makes intercepts of length ‘a’ on x-axis and that of ‘b’ on y-axis
then the eqn of line is : x/a + y/b = 1 …. The co-ordinates of point at which it intersects x-axis is (a,0)
and the point at which it intersects y-axis is (0,b)
(iv) Parametric Form : The eqn of straight line passing through the point (x1,y1) and making angle @
with the positive direction of x-axis is :
(x-x1)/cos@ = (y-y1)/sin@ = r , where ‘r’ is the distance of any point (x,y) on the line from (x1,y1)
That gives , x = x1 + rcos@ and y = y1 +rsin@ ( called parametric equations of a line)
(v) The eqn of straight line on which the length of the perpendicular from the origin O(0,0) is ‘p’ and
the perpendicular makes an angle @ with the x-axis is :
xcos@ + ysin@ = p ….. the co-ordinates of foot of the perpendicular are ( pcos@, psin@)
8. Angle between two lines :
(a) The angle @ between the two straight lines having slopes m1 and m2 is given by :
tan@ = ± { (m2-m1)/(1+m1m2)}
(i) If tan@>0.... @ is an acute angle.
(ii) If tan@<0 ….. @ is an obtuse angle
(iii) If lines are parallel then m1 = m2
(iv) If lines are perpendicular to each other then m1*m2 = -1
(b) The angle @ between the lines A1x + B1y + C = 0 and A2x + B2y + C = 0 is :
tan @ = (A1B2-A2B1)/(A1A2+B1B2)
(i) If lines are parallel then , A1B2 = A2B1
(ii) If lines perpendicular then A1A2 + B1B2 = 0
9. Perpendicular Distance :
(a) The length of a perpendicular from a point (x1,y1) to the straight line Ax + By + C = 0 is given by
|Ax1+By1+C|/rt(A^2+B^2) ….. i.e. the +ve value of Numerator is to be taken.
(b) Distance between parallel lines y=mx+c1 and y=mx+c2
= Difference between the distances of the lines y=mx+c1 and y=mx+c2 from the origin (0,0)
=(C1-C2)/rt(1+m^2) … where C1 & C2 are the distances of line y=mx+c1 and y=mx+c2 respectively
from origin.