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 line y=mx+b intersects the parabola y=x^2 at two points A and B. Find the point P on arc AOB that maximizes the area of triangle APB.
I want to see how you guys solve this problem. I wrote an equation for shortest distance between a point on the parabola and the line, then tried to...
Hi everyone.
I tried solving these questions but can't seems to get anywhere. I am not used to questions like these specially word problems. Sorry if this is the wrong section.
Question 1:
The receiver of the satellite dish is at the focus of the parabola dish. The focus is 80 cm from the...
Find the vertex, axis of symmetry, focus, and directrix of each parabola. Indicate whether the vertex is a max. or min. point.
Problem 45.
x=4y^2 - 6y + 15
Each pair of the following three lines cross at a point. Those points are the y-intercept, one of the x-intercepts, and the vertex of a parabola. Can you please explain to me how to find an equation for the parabola? And the other x- intercept?
y+8x=32
y+5x=32
y+3x=12
I have been trying...
I'm having problems solving this particular question
Using matrix calculation to find a general solution for a parabola that passes through points (x1,y1), (x2, y2), (x3, y3).
First I setted my equations
ax1^2+bx2+c=y1
ax2^2+bx2+c=y2
ax3^2+bx2+c=y3
But I've no idea on how I...
Hi,
I was giving a pub quiz type quesstion the other day, which I managed to solve without the use of formulae/mathematics, but it got me thinking about how to solve it mathematically too.
Imagine a chain stretched between two posts, that then droops in the middle, and forms a natural...
I've just finished my pure h/work but the final question has me a bit baffled because in lectures we've only been dealing with parabols where the focus is like (a,0) and the directrix is just y=k where k is a real number. However, for this question we've to find the eqt of the parabola whose...
I'm missing a step somewhere.
Without a calculator, graph y=3x^2-16x-12 by factoring and plotting zeros.
I have gotten as far as (x=-2/3) & (x=6) and know it's a parabola and pointing up because it has the positive x^2 so the graphing is easy enough, except...
The botb says the y-int is...
Hey!
I know that the horizonal and horizontal motion are unrelated, and the velocity remains the same (if there is no air resistance, friction etc)- this correlates with the whole 'shooting a gun and dropping a bullet at the same time, which one will hit the ground first? They will fall at...
Hey! I know that the horizonal and horizontal motion are unrelated, and the velocity remains the same (if there is no air resistance, friction etc)- this correlates with the whole 'shooting a gun and dropping a bullet at the same time, which one will hit the ground first? They will fall at the...
1)Find equations of both lines through point (2,-3) that are tangent to the parabola y=x^2+x.
2)The normal line to a curve c at a point Pis, by defininiton, the line that passes through p and is perpindicular to the tangent line c at P.
Where does the normal line to the parabola y=x^2-x...
find aan equation for he indicated parabola
1.focus(1,2), directrix x+y+1=0
2.vertex(2,0), directrix 2x-y=0
3.vertex(3,0), focus (0,1)
please tell me the steps how to find this 3 parabola equation...