Understand the concept of slope of a line and extend it to the concepts of conditions for parallelism and perpendicularity of lines and collinearity of three points
Let P(x, y) be any point on the line l having inclination . Let the line l meet the X-axis at A and the Y-axis at B.
Let Q() be a point on the line at a distance r from P(x, y), i.e., PQ = r
Now, QN = ML = OL − OM
= x − x1
Also, PN = PL − NL = PL − QM
= y − y1
In PQN, we have:
From (i) and (ii), we get:
This is the parametric form of equation of a line.
From the above equation, we have:
Thus, the coordinates of any point on the line at a distance r from the given point () are ().
Note: If Q() is a point on a line l which makes an angle with the positive direction of the x-axis, then there will be two points on line l at a distance r from Q() and their coordinates will be
The two points will be on the both sides of point Q.
Example: A line with inclination 60 is drawn through the point (1, 4). Find the coordinates of two points on this line which are at a distance of 2 units from the given point.
Here, (x1, y1) = (1, 4), and r = 2
We know that the parametric form of the equation is
Also, the coordinates of any point on the line at a distance 2 units from the given point (1, 4) are , i.e., .
Point of Intersection of Two Intersecting Lines
Theorem: If are two intersecting lines, then the coordinates of their point of intersection are
The equations of the two intersecting lines are as follows:
Solving the equations (i) and (ii), to eliminate gives
provided a1b2 − a2b1 ≠ 0 ...(iii)
Similarly to eliminate x, gives
provided a1b2 − a2b1 ≠ 0
Therefore, the coordinates of the point of intersection of lines (i) and (ii) are .
Remark: As the lines intersect, they are not parallel.
Note: By using Cramer's Rule (determinant form) as denoted by , we get:
From (iii) and (iv), the coordinates of the point of intersection are given by .
Example: Show that the lines and intersect at the point (4, 3).
The given equation are:
Multiplying equation (i) by 2 and solving, we get:
∴ y = 3
Substituting the value of y in equation (i), we get:
x = 3 + 1 = 4
∴ x = 4, y = 3
i.e., the point of intersection is (4, 3).
Thus, the lines and intersect at the point (4, 3).…
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