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.
The thing I have noticed is that parabola's angle from the x-axis keeps increasing...meaning the object is decreasing from that direction...if that is the case, then how does it go on in that direction forever even though, its speed going in that direction keeps decreasing?
At what rate...
Im In a 1st Year University math class and its been a while since i did this stuff and can't remember if I am doing this question right so any help would be much appreciated.
The question is to find all the info from the equation y=2(x-4)^2-3
I used the b+/- b^2 -4ac... equation and ended...
Im In a 1st Year University math class and its been a while since i did this stuff and can't remember if I am doing this question right so any help would be much appreciated.
The question is to find all the info from the equation y=2(x-4)^2-3
I used the b+/- b^2 -4ac... equation and ended...
Homework Statement
Find an equation of a parabola given three points without a vertex point.
Homework Equations
y=a(x-h)+k
The Attempt at a Solution
The parabola is upside down to I know a is negative.
Homework Statement
There is a point (4,2), from which I have to find the least distance to the parabola y^2=8x. I.e, the least distance connecting the point and the parabola.
Homework Equations
yy1=2a(x+x1)
Perpendicular distance = mod[(ax1+by1+c)/sqrt(a^2+b^2)]
The Attempt at a...
[SOLVED] Parabola Help
Homework Statement
Click me
Homework Equations
Unknown
The Attempt at a Solution
I can complete a) fine, by differentiating and sticking in y value to get m for tangent then -1/m for normal (-p). Then i used (y - y1) = m (x - x1) to provide normal equation.
I have...
Homework Statement
Triangle OAB is an isosceles triangle with vertex O at the origin and vertices A and B on the parabola y = 9-x^2
Express the area of the triangle as a function of the x-coordinate of A.Homework Equations
A = 1/2 bh
Distance formula (maybe)
Heron's Formula (an...
Homework Statement
Write the equation for the parabola. Vertex (0,0), axis along x-axis, passes thru (-2,-4).
The Attempt at a Solution
I thought since the parabola resides on the x-axis that I was supposed to use x^2=4py, with a parabola looking similar to this:
However, the...
Homework Statement
Find the focus of parabola: y:-(1/4)x^2 + 2x - 5
The Attempt at a Solution
Multiplying by 4 to get rid of fraction I get:
4y:-x^2 + 2x - 5
I will bring over the -5:
4y+5:-x^2 +2x
Completing Square I obtain:
4y+6:-(x-1)^2
Is all this correct thus far?
Homework Statement
Find the equation of the parabola if you have the following information:
Xintersects: (-4;0) and (0;0)
Yint: y = 7
Vertex: (-2;7)
Homework Equations
The equation must be in the standard form:
y=ax^2 + bx + c
The Attempt at a Solution
I know how to put it...
Homework Statement
A circle with a radius of 1inscribed in the parabola y=x^2, find the center of the circle. The figure shows the circle on the y-axis.
Homework Equations
y=x^2
r^2=(x-h)^2+(y-k)^2
The Attempt at a Solution
h=0. \frac{x^2+(y-k)^2}{r^2}=x^2
R^2=1 and the x^2 can be...
1. Homework Statement . The correct answer is E
2. Homework Equations :Procedure from our text:
"Step 1. Determine the differential equation for the given family F(x, y,C) = 0.
Step 2. Replace y' in that equation by −1/y'; the resulting equation is the differential equation
for the family of...
If someone could tell me how to go about it, I'd really appreciate it. Just point me in the right direction please. Kinda stuck.
The vertex of a parabola is (-3,1) and it touches the x-axis at the origin. Find the equation of the axis of the parabola.
Can somebody please help me with this question. I keep thinking I have made some progress on it and it turns out to be totally useless work. Spent almost 2 days struggling over this problem and 2 others and they have to be done for Monday. I have scanned in the question below...
alright so here is the problem.
The parabola y=x**2+3 has two tangents which pass through the point (0,–2). One is tangent to the to the parabola at (A,A2+3) and the other at (–A,A2+3). Find (the positive number) A.
i've attempted this problem several times but haven't been able to find the...
I've been playing around with parabolic mirror calculations in my free time, and right now I'm trying to derive the equation for the line that models the angle of reflection off of a point. I understand how to find the angle of incidence and that the reflection will have the same angle off the...
Using a graph to help you, find the equations of all lines through the origin tangent to the parabola
y=x^2-2x+4.
Sketch the lines on the graph.
I'm not sure how to start this off, I know the the first derivative would give me the slope of the tangent line at any point. If I find that...
Rotating water forms a parabola - why??
I've done some experiments involving rotating a beaker of water, and then measuring the height of the parabola that forms. I am now trying to explain this in my write up, but there are some things that I am simply not understanding, and any help would be...
A circle with the radius of 1 unit is inscribed in a parabola with the equation of y=x^2. Find the coordinate of the center of the circle.
The answer given in the book is (0, 5/4).
I attempt this question by letting the equation of the circle be x^2 + (y-c)^2 = 1, where c is the...
Hi I am stuck on the following:
Find the centre of a circle with a radius of 1 inscribed in the parabola y=x^2.
I am kinda stuck:grumpy:
http://img115.imageshack.us/my.php?image=csacacfu8.png
the white line is length=1
If I am given (y-2)2= 4 (x-3), how would I find the vertex, equation of the axis of symmetry, and the direction of the opening? I'm guessing that I have to use PF=PD, but it's confusing because it looks different from other things that I have done involving PF=PD.
Thanks.
Find the vertex, focus, and the directrix of the parabola.
I get tripped up sometimes, but I know how to find all of the stuff with an equation like this :
(x+1)^2 +8(y+3) = 0
But how do I find it with equations like this
y^2=-6x
or
x+y^2=0
Can you complete the square with...
The line y=mx+b intersects the parabola y=x^2 at points A and B. Find the point P on the arc AOB ofthe parabola (where O is the origin) that maximizes the area of the triangle PAB.
How do you find the equation of a parabola if you are given it's vertex and 1 point? For example, find the quadratic equation of a parabola that has a vertex of (-2,-2) and goes through the point (-1,0)
Let f(x) = 6-x^2 for 0<w<sqrroot(6); let A(w) be the area of the triangle formed by the coordinate axes and the line tangent to the graph of f.
see figure : http://img472.imageshack.us/img472/50/a5nt.jpg
Hey
I have a math problem that does not seem to work out. It is a word problem about a suspension bridge. The bridge's main cable is in the shape of a parabola, and the entire bridge, the road, is 200 ft long. The ends of the bridge, the longest virtical cables on the bridge, are 50 ft...
Hi guys, I started linear algebra just yesturday. I've read the first section of the first chapter and now I'm trying to do some problems from this section and I'm stuck on #7. I scanned the page from my math book (here's a link: http://img392.imageshack.us/img392/8476/lach11n72fb.jpg ). After...
Given {x | -50 < x < 50}, {y | 0 < y < 20} vertex at (0,20) it is a parabola find the equation in (x-h)^2=4p(y-k)
its pretty easy except how do i find p?
I am graphing a parabola for a calculus problem. I understand how you factor and use the two x values as your x-intercepts, but I'm not sure how this one would be graphed.
y=4x^2 - 25. I understand how the vertex of the parabola would be at negative 25, but I have no clue what to do woith...
I would like to verify my answer please:
x^2-2xy+y^2+5x+5y=0
using the formulas b^2-4ac=0 (indicates a parabola)
b^2-4ac<0 (indicates an ellipse)
b^2-4ac>0 (indicates a hyperbola)
2^2-4(1)(1)=0
4-4=0 therefore this graph must be a...
I need to find the equation of a parabola in standard form if the domain of an arch under a bridge is {-50 <= x <= 50} and the range is {0<=y<=20}
The span of the arch is 100 metres and the height is 20 metres.
standard form of a parabola is
(x-h)^2=4p(y-k)
i know the vertex is...
1. Determine the restrictions on the constants A, C and D so that Ax^2+Cy^2+Dx+Ey+F=0 represents a parabola that opens to the left.
I know that A=0, how do u know if CD>0 or CD<0...??
2. The equation Ax^2+By^2+Cy=0 represents an ellipse. If 0<A<B<C, then the vertices of the ellipse are on
a...
Feynman says that the path of projectile in a uniform gravitational field is a parabola, but the bottom of http://online.redwoods.cc.ca.us/instruct/darnold/CalcProj/Fall98/NateB/definition.htm says it's a cycloid. My calculation shows a parabola. Which is correct?
If parabola, why does a...
Can someone help me prove the following:
L=\mathop \lim\limits_{k\to \infty}\int_{-\frac{3}{4k}}^{\frac{3}{4k}} f(x)[-\frac{16k^3}{9}x^2+k]dx=f(0)
I'm pretty sure at the limit,
-\frac{16k^3}{9}x^2+k
becomes a delta function.
Essentially, it's that section of a narow parabola above...
Hi, I need to determine the equation of a parabola given the focus (2,3) and the directrix y=-1
I sketched out a parabola opening up wards with a vertex of (1,1)
I made two distance equations one for any point on the parabola to the focus, and one distance from the directrix to any point...
Determine the restrictions on the constants A, C and E such that the following equation is a parabola with a vertical axis of symmetry.
Ax^2+Cy^2+x+Ey=0
a) A=0, C cannot=0, E cannot=0
b) A=0, C cannot=0, E=0
c) A cannot=0, C=0, E=0
d) A cannot=0, C=0, E cannot=0
i know that A cannot=0...
I have a few problems for a review section in my book for a test comming up soon. i am confused on a few problems and was wondering if some1 could help please?
1.) The Axis is horizontal; the parabola passes through the points (0,4), (0,-1) and (6,1)
I am very confused on this problem...
I have tried to find some information of the expresions for a least-squares parabola coefficients (including their uncertaintities), then I have tried to do it for myself using the minimum condition for partial derivatives as same as with the least-squares line, but the expressions of coefs are...
:grumpy:
So i have this question which seems easy enough but maybe iam not thinking in the right mind frame or something.
The question is, find the equation of a parobola which has a curvature of 4 at the origin.
Some sort of hint/push in the right direction would be appreciated. Thanks