x – x1 = y – y1 = r
cos θ sin θ
where r is the distance of the point (x, y) on the line from the point (x1, y1)
Proof: Let the given line meets x – axis at T, y – axis at V and passes through the point A (x1, y1). Let P (x, y) be any point on the line at a distance r from Q (x1, y1) i. e. PA = r.
Draw PM perpendicular to OX, AN perpendicular to OX and AL perpendicular to PM. Then,
AL = NM = OM – ON = x – x1
and, PL = PM – LM = PM – AN = y – y1
In ΔPAL, we have
cos θ = AL/PA
cos θ = (x – x1)/r – (i)
and sin θ = PL/PA
sin θ = (y – y1)/r – (ii)
From (i) and (ii), we get
x – x1 = y – y1 = r
cos θ sin θ
This is the required equation of the line in the distance form.
Important Remarks
1. The equation of the line is
x – x1 = y – y1 = r
cos θ sin θ
x – x1 = r cos θ and y – y1 = r cos θ
x = x1 + r cos θ and y = y1 + r cos θ
Thus, the coordinates of any point on the line at a distance r from the given point (x1¬, y1) are (x1 + r cos θ, y1 + r sin θ). If P is on the right side of (x1, y1), then r is positive and if P is on the left side of (x1, y1), then r is negative. Since different values of r determine different points on the line, therefore the above form of the line is also called parametric form or symmetric form of a line.
2. In the above form one can determine the coordinates of any point on the line at a given distance from the given point through which it passes. At a given distance r from the point (x1, y1) on the line x – x1 = y – y1
cos θ sin θ
there are two points viz. (x1 + r cos θ, y1 + r sin θ) and
(x1 – r cos θ, y1 – r sin θ)
Now try it yourself! Should you still need any help, click here to schedule live online session with e Tutor!
About eAge Tutoring :
eAgeTutor.com is the premium online tutoring provider. Using materials developed by highly qualified educators and leading content developers, a team of top-notch software experts, and a group of passionate educators, eAgeTutor works to ensure the success and satisfaction of all of its students.
Contact us today to learn more about our guaranteed results and discuss how we can help make the dreams of the student in your life come true!