Verified answer

Conics analysis was devised by the Greeks to indicate quadratic curves. The conic is actually two right cones which meet each other at their mutual apex, and one cone is congruent to the other. We're not concerned with the depth of either cone; it need only be deep enough for the equation. The conic exists in x,y,z space and the apex is at (0,0,0), and the cones "open up" so that if you looked at the conic from "the top", you would see a circular shape on the x-y axis, and depth from the apex is measured in the + and - z direction. (sorry for the lengthy discription; I dont have the software to draw one and show it).

Now we take a plane and hypothetically "slice" through the conic, and consider the shape of the surviving conic portion in 2-d (x,y). If we slice on a horizontal plane across the conic, we would generate a circle. It's eqtn is

x^2 + y^2 = c^2 in rectangular coordinates, and r=c in polar coordinates. If we re-create the conic and slice at a "slant", we generate an ellipse with eqtn of ax^2+by^2=c^2. a and b are positive and a#b. The eccentricity of the ellipse describes the degree of "squishing" of the curve from a circle, where eccentricity>1. If the slice becomes parallel to the slant of the side of the conic, a parabola is generated. The vertex tells us how close the bottom of the parabola comes to the conic apex.

Now we turn our slicer so it is aligned parallel to the axis of the conic. We take a slice that cuts through the conic, but not deep enough to slice it in half. In this case, we get a regular hyperbola x^2-y^2=c^2, where c is a measure of the distance between the two parts of the curve. If we sliced directly on the axis of the conic, we would get two lines, which is a degenerate hyperbola (x^2=y^2). Now, if we sliced at an angle greater than the slant of the sides, we would cut off part of both parts of the conic and have a eccentric hyperbola x^2/a^2 -y^2/b^2 = c^2. I would venture that the directrix of the hyperbola is the line that is halfway between the two parts of the cuvre at their closest approach to each other.

Hope that helps.

Hi,

I am a Mathematics Tutor and have taken Pre-Calculus in the past with a grade of an A. From my understanding of a Conics, there are two ways to define a Conics: One way is to define it the way the Greeks did in the past by stating it in geometric terms such as the following:

AN intersection of a double-napped with a plane surface. This definition looks at a Conic from a pure geometric stance where a plane surface intersects either the upper || lower surfaces of the cone or the vertex. A plane that intersects the vertex of a Cone is said to be a degenerate Conics.

The 2nd way to look at && define a Conic would be merely the algebraic way in terms of xy variables with likable powers such as the following:

ax^2 + bxy + cy^2 + ex + fy + d = 0. Thus, a conic is an algebraic expression involving xy terms with 2nd degree powers.

The conic is a classification of several sets of planes && elliptical properties.

Hyperbolas, Parabolas, Circles, Ellipses are just several types of conics. Conics of the form ax^2 + bxy + cy^2 + ex + fy + d = 0 where both a && c are equal is called a Circle. If the product of ac > 0, then the conic is an ellipse. Otherwise, it is a hyperbola. A parabolic conic would entail the product of ac = 0.

In regard to your last question concerning the construction of conics involving the specified types, you must know the slight difference between just one of the two.

The difference between an Ellipse and a Hyperbola is that an Ellipse is a set of all points on a plane the sum of whose distance between fociis is a positive constans whereas a Hyperbola is a set of all points on a plane whose difference between the foci is a positive constant. Thus, the equation of an Ellipse is the sum of the ratio of the corresponding terms to their respective coefficients equal to the constant number 1. Likewise, the equation of a Hyperbola is the difference of the ratio of the xy terms to their respective coefficients. The equation of a Parabola involves the vertex formula (x-h)^2 + k with a standardized form of:

(x - h)^2 = 4p(y - k) with a directrix of y = k - p; the directrix is always perpendicular to the symmetric line of the parabola.

To understand the way an Ellipsis is formed, you need to understand the relationship between an Ellipsis && a Circle. An Ellipsis can be Circle iff the denominators of the terms are the same. Otherwise, the conic is an Ellipse. Also, the idea of eccentricity is unique to an ellipse because ellipse have eccentricity of c/a > 0 whereas Circles have no eccentricities. The eccentricity is merely the measure of the "ovalness" of a circular shape such as an Ellipse or hyperbola. Thus, the equations for an Ellipse && Hyperbola are the following:

Ellipse: x^2/a^2 + y^2 / b^2 = 1 with a conjugate axis of x being the the major focus.

Hyperbola: x^2/a^2 - y^2/b^2 = 1 with a transverse axis of x being the major focus. Also, the equation of a hyperbola can have interchanged axis such as the following:

y^2/a^2 - x^2/b^2 with the major axis of y and a minor of x.

In addition, the complete equation of a parabola in conic form would be constructed as the following:

(y - k)^2 = 4p(x - h)^2 where p is the focus and the directrix is at the line x = h - p; If both the parabola is symmetric with respect to the origin, then the parabola takes the following form:

y^2 = 4px || x^2 = 4py with a intermixing of the major and minor axis along with the directrix.

Tell me which one you prefer: Geometric || Algebraic explanation of Conics

Ohhyeah, That which I forgot to share is that polar form of conics. To make this one short and concise, you have to know the value for e. As in the previous post, if (e = 1), the polar form of the conic is a parabola with twice the directrix. Polar equations of conics take the form of y = ep/1 + ecostheta || y = ep / 1 + esintheta. There are several conditions that are associated with Conics: if (e > 1), the polar form is a hyperbola or otherwise an ellipse. Additionally, the expression (1 +/- ecostheta) is a horizontal shift, which is equivalent to the shift of an elementary function such as f(x + 1). But the sin is a vertical shifting, which is f(x) + 1. Also, they have their respective graphs, which are leminscates, rosecurves, && circles. But always take heed to the eccentricity and distance of the line and origin cause these are what determine the type of conic representing the polar form.

J.C

In polar coords the conics are given by r = [ e p ] / [ 1 + e cos Θ], the focus is at the origin and p = the distance from origin to the directrix, and for a parabola e = 1 & the directrix is twice the distance to the vertex...note in b) the directrix is x = 2 { r = 2 sec Θ}....finally the vertices are found when Θ = 0 or π and the axis of symmetry is the x axis

thank you FOR TAKING THE TIME to describe AND WRITE WITH finished PARENTHESES. First, enable's be conscious the 6 via multiplying it situations each and every exponent interior: (a^4b^9) / (a^12b^6) next subtract exponents of powers with comparable bases. a^4/a^a million = a^(4-12) = a^-8. b^9/b6 = b^(9-6) = b^3. a^-8 •b^3 or rewrite with all useful exponents as b^3 / a^8