In mathematics, a parabola is a plane curve which is mirror-symmetrical and is approximately U-shaped. It fits several superficially different mathematical descriptions, which can all be proved to define exactly the same curves.
One description of a parabola involves a point (the focus) and a line (the directrix). The focus does not lie on the directrix. The parabola is the locus of points in that plane that are equidistant from both the directrix and the focus. Another description of a parabola is as a conic section, created from the intersection of a right circular conical surface and a plane parallel to another plane that is tangential to the conical surface.The line perpendicular to the directrix and passing through the focus (that is, the line that splits the parabola through the middle) is called the "axis of symmetry". The point where the parabola intersects its axis of symmetry is called the "vertex" and is the point where the parabola is most sharply curved. The distance between the vertex and the focus, measured along the axis of symmetry, is the "focal length". The "latus rectum" is the chord of the parabola that is parallel to the directrix and passes through the focus. Parabolas can open up, down, left, right, or in some other arbitrary direction. Any parabola can be repositioned and rescaled to fit exactly on any other parabola—that is, all parabolas are geometrically similar.
Parabolas have the property that, if they are made of material that reflects light, then light that travels parallel to the axis of symmetry of a parabola and strikes its concave side is reflected to its focus, regardless of where on the parabola the reflection occurs. Conversely, light that originates from a point source at the focus is reflected into a parallel ("collimated") beam, leaving the parabola parallel to the axis of symmetry. The same effects occur with sound and other waves. This reflective property is the basis of many practical uses of parabolas.
The parabola has many important applications, from a parabolic antenna or parabolic microphone to automobile headlight reflectors and the design of ballistic missiles. It is frequently used in physics, engineering, and many other areas.
Problem statement : I copy and paste the problem as it appears in the text down below. I have only changed the symbol of the given acceleration from ##a\rightarrow a_0##, owing to its constancy.
Attempt : I must admit that I could proceed very little.
Given...
what i tried to do is to write y=v_0tsin alpha - 1/2gt^2 and x=v_0 cos alpha tand that t=x/v_0 cos alphai plug t in the formula for y and get that y= x tan alpha - gx^2/v_0^2 (tan^2 alpha -1)since jaan klada said there should be a quadratic equation (because its a parabola) i thought that...
Is it possible to trace the trajectory of an object using only its velocity and position, both of which are given as components. My method of doing so involves using the time until max height is reached, and using that time value to calculate the max height itself (h,k), then plugging in the...
This is not really a homework problem (it could be made to be though). I kind of made it up, inspired by a youtube math challenge problem involving parabolas, a water fountain where A = 1, R = 3, and H = 3. The solution given (h = 9/4) was based off simple math utilizing vertex form of a...
The general eqn of parabola is ##(x-h)^2=-4a(y-k)##. This is the parabola whose vertex doesn't lie on origin and axis is parallel to y axis. It opens downwards. Vertex is (h,k). What will be the focus of this parabola and what is ##a## in general form?
In the diagram a<0 which is...
Here I was going to use ##\int \vec F \cdot d\vec l = \frac{1}{2}mu^2##
What I got that is ##l=\frac{mu^2}{2eE}##. Here the question is what is ##l## (I took ##x## while doing the work but here I used ##l## instead of ##x##)? I was assuming that it's ##x## since I am calculating work in the...
Tangent is drawn at any point ( $x_1$ , $y_1$ ) other than vertex on the parabola $y^2$ = 4ax . If tangents are drawn from any point on this tangent to the circle $x^2$ + $y^2$ = $a^2$ such that all chords of contact pass through a fixed point ( $x_2$ , $y_2$ ) then
(A) $x_1$ , a , $x_2$ are in...
I have been puzzling over an equation that could be made to show the parabola of a projectile.
So far I have determined that the lateral and vertical velocities are needed, the lateral velocity should determine the x² function but after that I am stuck.
To specify I refused to look this up as...
Would the trivial solution be x=0,y=0?
Non trivial:
let y=x^2
\frac{dy}{dx}=2x, \frac{dx}{dy} = \frac12y^{-\frac12}
x=\frac14 y^{-\frac12}
here x=1 and y = 1/16 is a solution
but my book says the answer is x=1/2 and y=1/4
this is one answer that you get with the equation I derived, but I...
$\textbf{xy-plane}$ above shows one of the two points of intersection of the graphs of a linear function and and quadratic function.
The shown point of intersection has coordinates $\textbf{(v,w)}$ If the vertex of the graph of the quadratic function is at $\textbf{(4,19)}$,
what is the value of...
Hello, I am currently in my college holidays and I have decided to do some maths to improve. My weakness is graphing and I am hoping to get some help or the solution on this question.
Question:
Let P(k,k^2) be a point on the parabola y=x^2 with k>0.
Let O denote the origin.
Let A(0, a)denote...
Summary:: I have a upwards opening parabola where I know the Y vertex point = 0. I also know 2 arbitrary points separated by a step size on the parabola. How can I solve for x at the vertex point?
X=horizontal plane
Y=vertical plane
I have a upwards opening parabola where I know the Y vertex...
We have a circle (x^2 + y^2=2) and a parabola (x^2=y).
We put x^2 = y in the circle equation and we get y^+y-2=0. We get two values of y as y=1 and y=-2.
Y=1 gives us two intersection point i.e (1,1) and (-1,1). But y=-2 neither it lie on the circle nor on the parabola. The discriminant of the...
I am unsure how to go about this. I tried following the suggestion blindly and end up with with some cumbersome terms that are not the answer. From what I understand the derivative at each point would equal to T?
Answer: I just can seem to get to this. I think I'm there but can't get it in...
How did Archimedes discover the Quadrature of the parabola without the use of calculus?
If someone could please explain, I would be eternally grateful.
So I read a description saying something along the lines of, a Parabola does have a 2nd focus and directrix, but that they stretch off into infinity, whereas for the hyperbola the 2nd focus comes back round..?
Anyway, I'm trying to picture it and understand in relation to the eccentricity, e...
Summary: Find the equation of the parabola when the focus and the equation of tangent at the vertex is known.
Find the equation of the parabola when the focus is ##(0,0)## and the equation of tangent at the vertex is ##x - y + 1 = 0##.
General equation : ## ax^2 + by^2 + 2hxy + 2gx + 2fy + c...
Find the vertex, focus, and directrix of the following parabola:
y^2+12y+16x+68=0
The form we have been using is (y-k)^2=4p(x-h)
Any explanation would help too, I'm really stuck on this one.
Thank you!
Correct me if I'm wrong:
A parabola extends without limit toward parallel lines.
A hyperbola extends without limit toward diverging lines.
They have very different equations.
My question: is the former a specific instance of the latter?
Does a parabola = a hyperbola that happens to have...
Homework Statement
A disc of radius R rolls without slipping along the parabola y= ax2. Obtain the constrain equation
Homework Equations
Because there's no slipping, then:
##R d \theta = ds (1)##
Where ##\theta ## is the angle between the line from the center of the disc to a fixed point...
Homework Statement
You throw a ball straight upward with initial speed ##v_0##. How long does it take to return to your hand?
Homework Equations
##v=v_0+at##
##x-x_0=v_0t+\frac{1}{2}at^2##
The Attempt at a Solution
My question is a general one that relates to the described solution.
According...
Homework Statement
Find the value of z for which (10x-5)^2 + (10y-7)^2 = z^2((5x+12y+7)^2 is a parabola
Homework Equations
eccentricity of parabola=1
The Attempt at a Solution
I can solve this by expanding everything and writing h^2-ab=0 but this equation looks suspiciously similar to...
Homework Statement
A particle moves along a parabola on the x-y plane with equation ##y^{2}=2px## with constant speed ##1000m/s##.What is the magnitude of its acceleration?
Homework Equations
Parametric equations ##\vec{r}=(b^{2}t^{2}/(2p),bt)##.
The Attempt at a Solution...
I can't come up with this function for hours:
I just started to learn quadratic functions so there must be an obvious solution for this which I clearly missed...
<Moderator's note: Moved from a technical forum and thus no template. Effort in post #3.>
What is the condition of tangency of a line y=mx+c on parabola with vertex(h,k) ,say for parabola (y-k)2=4a(x-h)?
I could only find the condition of tangency on standard form of parabola, in the internet...
Homework Statement
##r=\frac 1 {cos(\theta)+1}##
y=-x
A region bounded by this curve and parabola is to be found.
2. The attempt at a solution
I have found the points of intersection but I am not sure what to do with the line (I need polar coordinates and it is not dependent on r :( )...
I have a quadratic regression model ##y = ax^2 + bx + c + \text{noise}##. I also have a prior distribution ##p(a,b,c) = p(a)p(b)p(c)##. What I need to calculate is the likelihood of the data given solely the extremum of the parabola (in my case a maximum) ##x_{max} = M = -\frac{b}{2a}##. What I...
In a graph , straight line intersects the parabola at(-3,9) & (1, 1) Then the equation is
A) x^2-2x+3=0
B) x^2+2x-3=0
C) x^2-3x+2=0
D) x^2-2x-3=0
I know that I can find the answer by substituting the known values to each options, but how to do it the proper way? We need at least three known...
Determine the reflection of a parabola y^2-2y-4x-11=0 by the line y = -x.
I know how to do it graphically, but please tell me how to do it algebraically.
Homework Statement
A partice is projected at an angle 60 degrees with the speed 10 m/s. Then latus rectum is ?
g= 10 m /s^2
Homework Equations
i calculated the maximum height.Now what??
The Attempt at a Solution
h= u u sin theta sin theta/2g
Homework Statement
The maximum range of a bullet fired from a toy pistol mounted on a car at rest is R0=40m .
What will be the acute angle of inclination of the pistol for maximum range when the car is moving in the direction of firing with uniform v=velocity 20m/s, on a horizontal surface ...
I have this problem to solve until tomorrow:
A point moves along the parabola r*cos^2(θ/2) = p/2, p > 0, in the direction that θ increases. At the time t=0, the point is on the verge of the parabola. The velocity is v = k*r, k>0.
What is the equation of movement, the radial acceleration and...
Prove that tangents to the focal cord of parabola are perpendicular using the reflection property of parabola ( A ray of light striking parallel to the focal plane goes through the focus, and a ray of light going through the focus goes parallel)
I don't know whether this is solvable with just...
Hey, I have this question I've been trying to figure out in an integration textbook. The part of the question that I'm having trouble understanding is basically this.
With the parabola below, find the x coordinate of A, if the line OA divides the shaded area into two equal parts.
The area of...
Homework Statement
Consider the curves: y = x^2 from 1/2 to 2 and y = \sqrt{x} from 1/4 to 4.
a. Explain why the lengths should be equal.
b. Set up integrals (with respect to x) that give the arc lengths of the curve segments. Use a substitution to show that one integral can be...
I want to plot an hormetic curve. This is the code I have so far:
\begin{tikzpicture}[scale = 0.75]
%preamble \usepackage{pgfplots}
\begin{axis}
[xlabel=Exposure,
ylabel=Benefit]
\end{axis}
\draw[black, line width = 0.50mm] plot[smooth,domain=0:6] (\x, {4-(\x-3)^2});
\end{tikzpicture}...
Homework Statement
the line goes through (0, 3/2) and is orthogonal to a tangent line to the part of parabola y = x^2, x > 0
Homework EquationsThe Attempt at a Solution
I have problems regarding finding the equation of tangent line to the part of parabola
because the question not specifically...
I made the problem up myself, so there might very well not be a rational answer that I like!
Homework Statement
A point-particle is released at height h0 is released into a parabola. The position of the particle is given by (x, y) and the acceleration due to gravity is g. All forms of friction...
Hi all. I need a bit of help determining the equation of some parabolas given points, intercepts and vertexes. Below are the exacts questions, any help will be much appreciated as I need this done soon!
1. A parabola has turning point (1,6) and passes through the point (-1,8). Find its...
Homework Statement
a) Show that for a given velocity V_0 a projectile can reach the same range R from two different angles \theta = 45 + \delta and \theta = 45 - \delta, as long as R doesn't go over the maximum range R_{max} = \frac{V_0^2}{g}. Calculate \delta in function of V_0 and R.
b)...
HI! I'm not sure if this can go in precalculus or not because I'm from Australia, and our Maths subjects don't get that specific until university level.
1. Homework Statement
For my assignment on quadratic functions, I have to find the equation (the the form of ax^2+bx+c) for a table of...
Let y = 2x^2 + 4x + \tfrac{7}{4} and line p a normal which go through the point (X, Y). Find the regions in xy-plane where there are either 3 distinct normals or only 1 normal.