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Conic Sections and Equations of the Second Degree

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Conic section can be defined as a curve that is obtained by Intersection (Cartesian product) of a cone with a plane. In analytic Geometry conic section can be defined as an algebraic curve that has degree 2. Here we will discuss conic sections and equations of the second degree. General equation of any conic section is given as:
Fp2 + Gpq + Hq2 + Ip + Jq + K = 0;
If value of F = 0 then we will get ‘F’ and ‘H’ in the equations. Different types of conic sections are Parabola, Circle, Ellipse and Hyperbola. Here we will discuss each conic section with help of a table mentioned below:

Name of conic section

Relationship of F and H

Parabola

F = 0 or H = 0 but both values of ‘F’ and ‘H’ will never be equals to 0.

Circle

In case of Circle value of ‘F’ and ‘H’ are equal.

Ellipse

In case of ellipse sign of ‘F’ and ‘H’ are same but ‘F’ and ‘H’ are not equal.

Hyperbola

In case of hyperbola signs of ‘F’ and ‘H’ are opposite.


Let’s see which equations are taken in second degree equation. In mathematics, generally we consider polynomial and quadratic equations in second degree equation. For example: Suppose we have an equation ax2 + bx + c = 0. Given equation is said to be Quadratic Equation because it has highest degree of 2 (square). To solve second order quadratic equation formula is defined which is given as:
X = -b + √ b2 – 4ac / 2a.
Circle, Parabola, Ellipse, hyperbola equation also has order of degree 2.
This is all about Conic sections and equations of the second degree.

Equations in Polar Coordinates

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Polar coordinate is a coordinate system in which each Point on coordinate plane is calculated using distance from a fixed point and an angle from fixed direction.

In Polar Coordinate Plane fix point is called pole and line segment from the pole in fixed direction is called as polar axis. Distance that is measured from pole is called as radial distance and angle is known as angular coordinate. If we measure angle Ɵ in counterclockwise direction of axis then it is known as positive angular coordinate. Angles in polar coordinate system are commonly denoted by degrees or radians.

If we talk about degrees in polar coordinate then we can say that degrees are used in navigation and surveying while radians are used in mathematics and in mathematical physics. In mathematical Geometry, polar axis is drawn along the horizontal axis.

Now we will see Equations in Polar Coordinates.
Suppose we have two polar coordinates and these two polar coordinates ‘s’ and ‘Ɵ’ can be converted to Cartesian coordinates ‘A’ and ’B’ using trigonometric function sine and cosine:
A = s cos Ɵ;
B = s sin Ɵ;

If ‘s’ is greater than equals to zero than Cartesian coordinates A and B can be converted to polar coordinate in interval (-π, π) by pythagoras theorem. Using Pythagoras theorem polar coordinate equation can be written as:
u = √ (p2 + q2); and angle Ɵ is equals to:
Ɵ = a tan2 (q, p) where atan2 is a common variation on arctangent function.
We can also write polar Equation of a curve which is:
Now see the case of Circle: Equation of circle is given as:
R = r0 cos (Ө - ∂) + √ a2 – r20 sin2 (Ө - ∂), in this way we can also find equation of line, conic section etc.

Transformation to Rectangular Coordinates

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A Point in mathematics can be represented using two representations namely rectangular or Cartesian and polar coordinate form. Later one is described in terms of angle that specifies the direction of vector and a particular distance from a fixed point representing the magnitude of vector. Let there be a point whose polar form is given as:
Q (n cos r, n sin r)
Where, 'n' is the distance from origin and 'r' represents the angle made by line having coordinates Q (n cos r, n sin r) and A (0, 0).

For transformation to rectangular coordinates or Cartesian coordinates we first need to memorize the following representation of rectangular coordinates in polar form given as:
X = n cos r........ equation 1
Y = n sin r............ equation 2
Where, n ≥ 0 & 0 ≤ r < 2 pi.
So, to convert the Polar Coordinates to corresponding rectangular coordinates we need to substitute the values of x and y- coordinates in polar form to get X and Y. X represents the x – coordinate in x – y plane and Y represents the y - coordinate in the x – y plane. Let us consider an example to understand this conversion better:

Example: Polar coordinates of some point lying in space is given as (r, angle) = R (25, 450). Find out its equivalent rectangular coordinates.

Solution: According to the given information we can write the coordinates in polar form as:
R (25 cos 450, 25 sin 450)
Using equations 1 an 2 we can write:
X = 25 cos 450
Or X = 25 / 2 ½
Y = 25 sin 450
Or Y = 25 / 2 ½
Thus rectangular or Cartesian representation of the point R (25 cos 450, 25 sin 450) can be written as P (25 / 2 ½, 25 / 2 ½).

Simplification of the Equation in Rectangular Coordinates

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Simplification of the equation in rectangular coordinates is similar to solving two Linear Equations with two unknown variables. Rectangular coordinates are nothing but the points that lie on the line whose equation in general is written for all such points. In fact general equation of line is given in the form of rectangular coordinates as:
Y = p X + D,

Where, 'p' represents the Slope of line and 'D' denotes the intercept made by the line on y – axis. We will see that solution of two linear equations is their Intersection Point, which can also be represented in Cartesian or rectangular form i.e. x- coordinate along x - axis and y- coordinate along y - axis. Suppose that there are two linear equations given as follows:
16 x + 15 y = 100 and 8 x + 7 y = 50. To solve these linear equations given in rectangular coordinates form we have to apply several arithmetic operations. We first multiply the second equation 8 x + 7 y = 50 by 2 and then subtract it from first equation 16 x + 15 y = 100 to get:

(16 x + 15 y = 100) – 2 * (8 x + 7 y = 50) = (16 x + 15 y = 100) – (16 x + 14 y = 100),
Solving it for 'y' we get:
(y = 0),

Substituting the value of y in any of the two linear equations: Suppose we choose the equation 8 x + 7 y = 50 and substitute y = 0 we get the value of x as follows:
8 x + 7 y = 50,
8 x + 7 (0) = 50,
8 x = 50,
x = 50 / 8,
x = 25 /4.