GENERAL EQUATIONS OF CONIC SECTIONS

General Equation of a conic section is of the form: 

Ax² + Bxy + Cy² + Dx + Ey + F = 0

(The letters A-F are constants and the "=" sign could also be replaced with an inequality sign.) This general equation can be transformed into different specific equations with the form of that equation dictated by the actual type of quadratic relation.

Circle

A circle is a collection of points (x,y) in a coordinate plane, such that each point is equidistant from a fixed point (h,k) known as the center. For circles the coefficients for the x2 and y2 terms in the general quadratic relationship are equal (i.e. A=C).

The equation of a circle in standard form is as follows:              (x-h)² + (y-k)² = r²

Remember:

• (h,k) is the center point.
• r is the radius from the center to the circle's (x,y) coordinates.

Example: x² + y² + 6x - 4y - 12 = 0

Step 1 - Commute and associate the x and y terms; additive inverse the -12:

(x² + 6x) + (y² - 4y) = 12

Step 2 - Complete the squares, (what you do to one side be sure to do to the other side):

(x² + 6x + 9) + (y² - 4y + 4) = 12 + 9 + 4

Step 3 - Factor: (x + 3)² + (y - 2)² = 25 = 5²

Observations

1. The conic section will be a circle since the x² and y² terms have the same sign and equal coefficients.
2. The center, (h,k), is (-3,2). Note, these are the values of x and y which make the corresponding term equal to zero.
3. The radius of the circle will be 5 units, since the square root of 25 is 5.
4. A circle can be drawn with a compass or one thumbtack and a string.
5. The eccentricity of a circle is zero (e=0).

Ellipse

An ellipse is also a collection of points (x,y) in a coordinate plane. It is very similar to a circle, but somewhat "out of round" or oval. For an ellipse, the x2 and y2 terms have unequal coefficients, but the same sign (AC > 0). (The plural of ellipse is ellipses, which is also: .... Both stem from the same basic root meaning to leave out.)

Ellipses have the following standard form:            

((x-h)/r_x)² + ((y-k)/r_y)² = 1

Remember:

• (h,k) is the center point.
• r_x is the length of the radius in the ± x-direction.
• r_y is the length of the radius in the ± y-direction.

Example: x² + 4y² = 16

Step 1 - Divide both sides by 16:                 

(x²)/16 + (4y²)/16 = 1.

Step 2 - Simplify the second term: (x²)/16 + (y²)/4 = 1.

Step 3 - Factor/rewrite in standard form: (x/4)² + (y/2)² = 1.

Observations

1. The SAME sign but DIFFERENT coefficients for the x² and y² terms tell us that the graph will be an ellipse.
2. The two denominators, 4 and 2, (located in step 3) tell us that the ellipse's vertices are found 4 units from the center (0,0) in the ± x-direction and two other critical points are located 2 units from the center in the ±y-direction.
3. r_x=4 is called the x-radius and is the distance from the center to the ellipse in the x-direction.
4. r_y=2 is called the y-radius and is the distance from the center to the ellipse in the y-direction.
5. The semi-major axis is the larger of r_x and r_y, in this case 4.
6. The semi-minor axis is the smaller of r_x and r_y, in this case 2.
7. Semi- means half. Thus the major and minor axes are twice the semi-major and semi-minor axes.
8. An ellipse can also be described as the set of points in a plane such that the sum of each point's distance, d₁ + d₂, from two fixed points F₁ and F₂ is constant. Thus an ellipse may be drawn using two thumbtacks and a string.
9. F₁ and F₂ are foci, that is each is a focus. They are located at (h±c,k) or (h,k±c)
10. The distance from the center to a focus is the focal radius.
11. If a is the semi-major axis and b is the semi-minor axis, then c is the focal radius, where d₁ + d₂ = 2a, and c²=a²-b². In this case, c²=16-4=12.
12. The eccentricity e of an ellipse is given by the ratio: e=c/a. Since c < a and both are positive this will be between 0 and 1. An eccentricity close to zero corresponds to an ellipse shaped like a circle, whereas an eccentricity close to one corresponds more to a cigar.
13. The latus recta of an ellipse are line segments through a focus with endpoints on the ellipse and perpendicular to the major axis. Their length is 2b²/a.

Parabolas

A Parabola has an equation that contains only one squared term. If the x2 term is excluded, then the graph will open in an x-direction. If the y2 term is excluded, then the graph will open in a y-direction. Only graphs which open in the ±y-direction are quadratic functions, thus those which open in the ±x-direction are quadratic relations.

Parabolic functions have the general equation: y = ax² + bx + c

Example:          x = -2y² +12y -10

Observations

1. The conic section will be a parabola because there is only one squared term, y².
2. Since the x² term is missing, the graph will open in an x-direction, specifically the -x since C < 0.
3. The y-coordinate of the vertex is found by the formula: k = -b/2a. So that k = -12/2(-2) = 3.
4. The x-coordinate of the vertex is:    h = -2(3²) + 12(3) - 10 = 8.
5. The eccentricity of a parabola is one (e=1).
6. A parabola can be described as the set of coplanar points each of which is the same distance from a fixed focus as it is from a fixed straight line called the directrix.
7. The midpoint between the focus and the directrix is the vertex. The line passing through the focus and the vertex is the axis of the parabola.
8. A focal chord is a line segment passing through the focus with endpoints on the parabola.
9. The latus rectum is the focal chord perpendicular to the axis of the parabola.
10. Another standard form for a parabola is:  (x-h)²=4p(y-k) or    (y-k)²=4p(x-h)
11. The focus lies on the axis p units from the vertex: (h, k+p) or (h+p,k).
12. The directrix is the line y=k-p or x=h-p

Hyperbolas

An Hyperbola has two symmetric, disconnected branches. Each branch approaches diagonal asymptotes*. Hyperbolas can be detected by the opposite signs of the x2 and y2 terms. (AC < 0).

Hyperbolas have the specific equations:

((x-h)/r_x)² - ((y-k)/r_y)² = 1     OR     -((x-h)/r_x)² + ((y-k)/r_y)² = 1

If the sign before the x² term is POSITIVE (A > 0), the hyperbola will open toward the ±x-direction. But if the sign before the y² term is POSITIVE (C > 0), the hyperbola will open toward the ±y-direction.

Remember:

• (h,k) is the center point.
• r_x is the distance from the center to the hyperbola's ± x-direction vertex (or asymptote).
• r_y is the distance from the center to the hyperbola's ± y-direction vertex (or asymptote).
• asymptotes have slopes of r_y/r_x and -(r_y/r_x)
• A hyperbola is the set of points in a plane such that for each point (x,y) on the hyperbola, the difference between its distance from two fixed foci is a constant.
• The semi-major axis, a, is the larger of r_x and r_y.
• The semi-minor axis, b is the smaller of r_x and r_y.
• The transverse axis connects the two vertices.
• The conjugate axis is perpendicular to the transverse axis.
• c²=a²+b².
• The eccentricity e of a hyperbola is given by the ratio: e=c/a. Since c > a and both are positive this will be greater than 1. If e is close to one, the hyperbola will be narrow and pointed; whereas if e is large, the hyperbola will be nearly flat.

Example: -(x/4)² + (y/3)² = 1

Observations:

1. The conic section will be a hyperbola since the x² and y² terms have different signs.
2. The graphs open in the ±y-direction since the sign before the y-term is positive.
3. The asymptotes would have a slope of 3/4 or -(3/4).

SUBMITTED BY: PAWAN KUMAR JANGRA.

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