1.10 Further exercises
Determine the equation of the line through each of the following pairs of points. Show that both equations can be written in the form ax + by = c, for some real numbers a, b and c, where a and b are not both zero.
(a) (−2, −4) and (1, 6).
(b) (0, 0) and (7, 3).
(a) Since (−2, −4) and (1, 6) lie on the line, its gradient is
It follows that the equation of the line is
which can be simplified to
10x − 3y = −8.
This equation is of the desired form, with a = 10, b = −3 and c = −8. (Any multiple of these numbers is also a valid answer.)
(b) Since the line passes through the origin and the point (7, 3), it has an equation of the form y = mx, for some m. The coordinates of (7, 3) must satisfy the equation y = mx. Thus 3 = 7m, so
Hence the equation of the line is
This can be written as
3x − 7y − 0,
which is of the desired form, with a = 3, b = −7 and c = 0.
Determine the values of k for which the lines 3x + 4y + 7 = 0 and 2x + ky = 3 are
- (a) parallel,
- (b) perpendicular.
The gradients of the lines 3 x + 4y + 7 = 0 and 2 x + ky = 3 are and , respectively.
Thus the lines are
(a) parallel if , that is, ;
(b) perpendicular if , that is, .
Sketch the lines with the following equations, on a single diagram:
Determine the coordinates of the points of intersection of the lines in Exercise 14.
Let A be the point of intersection of the lines y = −3x and . We equate the two expressions for y to obtain
Multiplying through by 3 gives
−9x = x + 6,
Since A lies on the line y = −3x, it follows that
So the point A has coordinates
Next, let B be the point of intersection of the lines and y − 3 = 3(x − 3). We rewrite the second equation, and equate the two expressions for y to obtain
Multiplying through by 3 and collecting terms gives
x = 3.
Since B lies on the line it follows that y = 3. So the point B has coordinates (3, 3).
Finally, let C be the point of intersection of the lines y = −3x and y − 3 = 3(x − 3). We rewrite the second equation, and equate the two expressions for y to obtain
−3x = 3(x − 3) + 3.
Collecting terms gives
6x = 6,
x = 1.
Since C lies on the line y = −3x, it follows that y = −3. So the point C has coordinates (1, −3).
Determine whether each of the pairs of planes given by the following equations intersect, are parallel, or coincide.
(a) x = 1 and y = 2.
(b) z = 1 and z = 3.
Illustrate your answer to each part with a sketch.
(a) The planes with equations x = 1 and y = 2 are parallel to the (y, z)-plane and the (x, z)-plane, respectively. They intersect in a line, as shown.
(b) The planes with equations z = 1 and z = 3 are both parallel to the (x, y)-plane. They are parallel to each other, as shown.
Determine the distance between the points (1, −2, 3) and (−2, 3, −1) in 3.
We use the Distance Formula given in Section 1.9. The required distance is thus