Concurrent Lines: Three or more straight lines are called concurrent lines if they meet in a point,
(i) Find the point of intersection of any two lines and show that it also satisfy the equation of the third line.
(ii) If l = 0, m = 0 and n = 0 are equations of the given lines and if for any three constants p, q, r not all zero to be found by inspection lp + mq + nr = 0 takes the form 0x + 0y + 0 = 0, then the three lines are concurent.
In particular, the lines l = 0, m = 0 and n = 0 are concurrent if l + m + n = 0.
(iii) The lines a1x + b1y + c1 = 0, a2x + b2y + c2 = 0 and a3x + b3y+ c3 = 0
are concurrent, if = 0
Distance between Parallel Lines:
(i) Choose a convenient point on any of the lines (put x = 0 and find the value of y or put y = 0 and find the value of x). Now the perpendicular distance from this point on the other line will give the required distance between the given parallel lines.
(ii) Write the equation of two parallel lines a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0 in such a way that a, and a2 are of the same sign. Then two lines will lie on the same side of the origin if c1 and c2 are of the same sign and on opposite sides of the origin if c1 and c2 are of opposite sign.
(iii) Find the perpendicular distance P1 and P2 of both the lines from origin and with their signs, the required distance between the parallel lines is P1 - P2, i.e.,
Note: First method gives the answer very quickly.
Position of a Point with respect to a Line: Let the given line be ax + by + c = ° and observing point is (x1, y1), then
(i) If the same sign found by putting in equation line of x = x1, y = y1 and x = 0, y = 0, then the point (x1, y1) is situated on the side of origin.
(ii) If the opposite sign found by putting in equation of line x = x1, y1 = y1 and x = 0, y = 0, then the point (x1 y1) situated opposite side to origin of the line.
Orthocentre: The point of intersection of perpendiculars drawn from vertices of a triangle on opposite sides is called the orthocentre of triangle. The orthocenre of a right angled triangle is the point at which the right angle form.
Area of a Parallelogram: Area = where P1 and P2 are the distances between parallel sides and q is the angle between two adjacent sides.
In case of a rhombus p1 = p2, Also area of rhombus = (1/2) d1d2 where d, d2 are the lengths of two perpendicular diagonals of a rhombus as in figures.
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