Find the equation of a line perpendicular to: a the line x+ 3y = 0 and passing through point (-3, 2) b the line y =-3x+ 0.75 and passing through (1, 1.5).
Family of Curves
A family of curves is a group of curves that are each described by a parametrization in which one or more variables are parameters. In general, the parameters have more complexity on the assembly of the curve than an ordinary linear transformation. These families appear commonly in the solution of differential equations. When a constant of integration is added, it is normally modified algebraically until it no longer replicates a plain linear transformation. The order of a differential equation depends on how many uncertain variables appear in the corresponding curve. The order of the differential equation acquired is two if two unknown variables exist in an equation belonging to this family.
XZ Plane
In order to understand XZ plane, it's helpful to understand two-dimensional and three-dimensional spaces. To plot a point on a plane, two numbers are needed, and these two numbers in the plane can be represented as an ordered pair (a,b) where a and b are real numbers and a is the horizontal coordinate and b is the vertical coordinate. This type of plane is called two-dimensional and it contains two perpendicular axes, the horizontal axis, and the vertical axis.
Euclidean Geometry
Geometry is the branch of mathematics that deals with flat surfaces like lines, angles, points, two-dimensional figures, etc. In Euclidean geometry, one studies the geometrical shapes that rely on different theorems and axioms. This (pure mathematics) geometry was introduced by the Greek mathematician Euclid, and that is why it is called Euclidean geometry. Euclid explained this in his book named 'elements'. Euclid's method in Euclidean geometry involves handling a small group of innately captivate axioms and incorporating many of these other propositions. The elements written by Euclid are the fundamentals for the study of geometry from a modern mathematical perspective. Elements comprise Euclidean theories, postulates, axioms, construction, and mathematical proofs of propositions.
Lines and Angles
In a two-dimensional plane, a line is simply a figure that joins two points. Usually, lines are used for presenting objects that are straight in shape and have minimal depth or width.
![8 Find the equation of a line perpendicular to:
a the line x+ 3y = 0 and passing through
point (-3, 2)
b the line y = –3x + 0.75 and passing
through (1,1.5).
9 Three farmhouses are to be powered by
wind. The farms are located at A(60, 20),
B(220, 120) and C(240,40), where the
coordinates are in metres. A wind turbine
is to be placed at the point equidistant
from the farms A, B and C. Find:
a i the equation of the line (AB)
ii the midpoint of line segment [AB]
iii the equation of the perpendicular
bisector of line segment [AB]
bi the equation of the line (AC)
ii the midpoint of line segment [AC]
ii the equation of the perpendicular
bisector of line segment [AC]
ci the equation of the line (BC)
ii the midpoint of line segment [BC]
iii the equation of the perpendicular
bisector of line segment [BC].
d Determine the coordinates of the point
T that is equidistant from the farms A,
B and C.
The blade of a wind turbine is 25 m long.
Current regulations require that the distance
between the turbine and a house be at least
three times the length of the blade.
e Determine whether the wind
turbine meets current regulations for
installation at point T.
f Determine the area that one wind
turbine needs to function. Give your
answer to the nearest integer.
10 The map of a rectangular province is
shown with the positions of four landing
platforms for helicopters. The points A,
B, C and D, which represent the landing
platforms, are placed on a coordinate
grid centred at one of the corners of the
province. (Each unit along a coordinate
axis is 100 miles.)
Emergency medical teams are flown to
any point in the province as needed.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7682c2dd-32fe-40c3-bea4-e2ab46c62205%2F96074835-2d67-4567-9283-908aada86ba7%2Fczk99yg_processed.png&w=3840&q=75)
![When an emergency occurs, the medical
team that is based closest to the location of
the emergency will fly out to the location.
y
8
7
B
6
D
4
3
JA
2
1
1 2 3
4
6 7
8
9 10
a Construct a Voronoi diagram for the
four sites A, B, C and D.
A neighbouring province asks whether
the medical team based at point B(2,6)
can also help out in their province. The
director replies that it would be possible
if the team is currently covering a smaller
area than at least two of the other medical
teams.
b Find where the perpendicular bisector
of [AD] meets the line y= 4.
c Find the area of the cells surrounding:
i A
ii B ii c
iv D.
Hence state whether or not the medical
team based at A will be able to support
the other province.
Exam-style questions
11 P1: The point M is the midpoint of points
A and B. If A (4, –6, 10) and M (7,3, –5),
find the coordinates of the point B.
(5 marks)
12 P1: The line L has equation y = 3x – 5.
For the lines given below, state with
reasons whether they are parallel to L,
perpendicular to L, or neither. (6 marks)
i y=-x-7
3
ii
-6x + 2y + 8 = 0
iii y – 5 = 2(x – 7) iv y =x+4
v x+ 3y + 9 = 0](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7682c2dd-32fe-40c3-bea4-e2ab46c62205%2F96074835-2d67-4567-9283-908aada86ba7%2F9jwcqa_processed.png&w=3840&q=75)
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