Chapter 1: Differentiation; Section 2: Real Functions; page 5
Functions and Geometry, Exercise 3
Question
The centre of a circle with radius r is at the origin of the coordinate system. A straight line is given in its general form (1.2.5.12).
What is the condition for the line not to have any common points with the circle, and what are the closest points of the line, and of the circle in this case?
What is the condition for the line to have only one common point with the circle, and what is this point?
What is the condition for the line to intersect the circle, and what is the length of the shortest arc obtained?
(1.2.5.12)
Reminder
The distance d between a point
and a line expressed in its general form (1.2.5.12) was calculated in the previous exercise (2):
From the same exercise the coordinates of the line closest to the point are:
Parts 1-4
These parts are related to the question 1.
The centre of the circle =(0,0) is at distance d from the line (1.2.5.12). According to the the previous exercise (see reminder above):
The condition for the line not to have any common point with the circle is d>r or according to the part 1:
The closest point of the line to the circle is according to the previous exercise (see reminder above) is:
The closest point of the circle to the line could be scaled from the point found in part 3, by the factor:
as explained in the note below.One obtains therefore:
Parts 5-6
These parts are related to the question 2.
The condition is d=r, and according to part 1:
According to parts 3 and 5, the common point is:
Parts 7-9
These parts are related to the question 3.
The condition is
and according to part 1:
If we denote by β the angle between the two radii pointing to the intersections with the line, then the length of the arc l should be:
This angle is related to d and r by:
and therefore one obtains:
Note about part 4
The closest point of the circle lies on a line between the point from part 3 and the centre of the circle, which is at the origin of the coordinate system. For this reason the slope of this line
is the ratio of the coordinates
which remains constant for any point of this line.