In mathematics, a conic section (or simply conic) is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a special case of the ellipse, though historically it was sometimes called a fourth type. The ancient Greek mathematicians studied conic sections, culminating around 200 BC with Apollonius of Perga's systematic work on their properties.
The conic sections in the Euclidean plane have various distinguishing properties, many of which can be used as alternative definitions. One such property defines a non-circular conic to be the set of those points whose distances to some particular point, called a focus, and some particular line, called a directrix, are in a fixed ratio, called the eccentricity. The type of conic is determined by the value of the eccentricity. In analytic geometry, a conic may be defined as a plane algebraic curve of degree 2; that is, as the set of points whose coordinates satisfy a quadratic equation in two variables, which may be written in matrix form. This equation allows deducing and expressing algebraically the geometric properties of conic sections.
In the Euclidean plane, the three types of conic sections appear quite different, but share many properties. By extending the Euclidean plane to include a line at infinity, obtaining a projective plane, the apparent difference vanishes: the branches of a hyperbola meet in two points at infinity, making it a single closed curve; and the two ends of a parabola meet to make it a closed curve tangent to the line at infinity. Further extension, by expanding the real coordinates to admit complex coordinates, provides the means to see this unification algebraically.
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.
I'm doing a question in my book on conics where there is a circle cutting through a parabola. There are three points S - the focus P - a point on the parabola and Q - the point where the tangent at P meets the directrix.
The focus is at S[1,0] and the dirextrix is x=-1
point P is [t^2, 2t]...
I have question for you,How calculate closest distance between two
ellipse in space.
orbit first : r = (p1)/(1+epsilon1*cos(theta1))
second orbit : r = (p2)/(1+epsilon2*cos(theta2))
the relation between two ellipse is some euler angles call them
first angle :a1
second...
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...
I think that I'm over looking something with this problem. Below is the equation of an hyperbola in polar form.
R=\frac{1}{1 + 2cos{\theta}}
when \theta =\pi shouldn't R = -1? And not R= 1
Am I over looking some property of the \cos function?
Even when i evalute this expression at...
Your task is to design a curved arch similar to the a tunnel for cars. with a horizontal span of 100 m and a maximum height of 20 m.
Using a domain of {x:-50<=x<=50} and {y:0<=y<=20} determine the following types of equations that could be used to model the curved arch.
the equation of a...
I have been taught that the equation Ax^2 + Cy^2 +Dx + Ey +F = 0 represents a general form of conics.
Then the conditions of the coefficients in the equation could identify which type of conics the equation represents...
Circle: A=C
Ellipse: A does not=C and AC>0
Hyperbola: AC<0 and...
6x^2 + 2y^2 - 9x +14y -68=0
a) which conic is represented by the equation why?
I think the ellipse is represented by the equation because a does not = b and ab >0
b)What value of "a" would transform the conic into a circle?
I think when a=b and ab>0 then the conic will be...
A flash light is pointed at a wall so that the angle between the beam and the wall is 65 degrees
a) which conic section is produced? I am not sure if this is asking for the shape but I think the answer is an ellipse
b) How would you adjuct the angle of the beam to produce a circle on the...
Problem I:
(The coeffiecients throw me off, I don't know what I'm supposed to do with them)
9x^2 + 16y^2 = 144
Determine:
a) coodinates of the centre
b) lengths of the major and minor axes
Problem II:
Sketch a graph of the ellipse
4x^2 + (y+1)^2 = 9
PS: For...
Ok I seem to be having problems with changing the general form of a conic to standard form. I'm mainly confused with how to factor, since I haven't done it in a while, as well as how to go about completing the square.
Here's one of my problems:
2x^2 + y^2 + 12x – 2y + 15=0
I rearranged...
"A line segment through a focus with endpoints on the ellipse and perpendicular to the major axis is a latus rectum of the ellipse. Therefore, an ellipse has two latus recta. Show that the length of each latus rectum is 2b^2/a."
I've been stuck on this for a little while now. Can anyone point...