2. Find the distance between the parallel lines: L: 2x- 5 y + 3 = 0 ; (Hint: It suffices to compute the distance from any particular point P, on L, to the line La.) La: 2 x- 5 y + 7 = 0.
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.
See handout. #2 Find the distance between the paralle lines.
![signment 11
Due date : Friday, December 11
Section 22.4, Page 707
#1, 2, 6*, 7*, 14, 15, 17.
*Note: The vertex is at the origin in these problems.
Required Additional Exercises
Exercises #1 – 5 (below).
1. Find the distance from the point to the given line by using the distance theorem
http://mypages.iit.edu/~maslanka/PointtoLine.pdf
(a) 2 x – 4 y + 2 = 0 ; (1,3)
(b) 4х+5у-3-0; (-2,4)
2. Find the distance between the parallel lines:
L: 2 x- 5 y +3 = 0; L2: 2 x – 5 y + 7 = 0.
(Hint: It suffices to compute the distance from any particular point P, on L to the line La.)
3. Find the equation of the line L bisecting the angle from Li to La given
Li: 3 x – 4 y – 2 = 0 ;
Hint: Note that if P = ( x , y) is any point on the bisector L then its coordinates satisfy
L2: 4 x – 3 y + 4 = 0.
the condition:
|A,x + B1y+ C¡| _ |A,x + B,y+ C¿|
VA,? + B,2
VA,² + B,²
|A,² + B,² · ( Aµx + B¡y+ C;) = ± VA,² + B,² · (A,x + B2y+ C,) (*)
where L: A,x+B,y +C, =o and L2: A,x + B,y + C, = 0 .
The solutions to (*) yield the equations of both lines bisecting the angles between L, and L2.
4. Find the two points of intersection of the circles:
x² + y 2 + 5 x + y – 26 = 0 ; x² + y² + 2 x – y – 15 = 0.
5. Find the equation of the parabola P with focus F = (1,1) and the directrix
D: x + y = 0 in two ways:
(a) By using the fact that : Q = ( x , y ) on P → d(Q,F ) = d ( Q , D ).
Hint: Square both sides of your equation for the parabola in order to eliminate the
square roots and absolute values and then simplify it.
(b) By rotating the parabola y² =2 z x by 45° about its axis and translating
1
its graph + units horizontally and +; units vertically.
Hint: Transform the equation of the parabola to polar coordinates before rotating
this curve. Then transfer back to xy-coordinates in order to translate the
equation of the rotated parabola. Refer to the handout:
http://mypages.iit.edu/~maslanka/R&T_Thrms.pdf for details on rotation
and translations theorems.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe31eb4c2-4ccb-4545-9098-1418f9db9f92%2Fbf863036-cfa2-49f1-8851-8a067618fa39%2F51xsrtr_processed.png&w=3840&q=75)
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