INTRODUCTION TO CONIC SECTIONS

A conic section is a curve formed by intersecting a plane with a double-napped cone, which has two opposed ends and extends infinitely. The plane's angle of intersection with the cone determines the shape of the conic section. If the plane cuts parallel to the cone's base, it forms a circle. When a plane cuts the cone at an angle but does not intersect the base, it forms an ellipse. If the plane is angled such that it cuts parallel to the side of the cone, the intersection forms a parabola. Finally, if the plane cuts both nappes of the cone, but at an angle steeper than that of the side, a hyperbola is created. These curves—circles, ellipses, parabolas, and hyperbolas—are collectively known as conic sections. Each conic has unique geometric properties and can be defined by specific equations in a coordinate plane, based on the position and orientation of the cone and the intersecting plane.

It is circle when the plane is Horizontal.

It is ellipse when the tilted plane intersects only one cone to form a bound curve.

It is parabola when the plane intersects only one cone to form an unbounded curve.

It is hyperbola when the plane (not necessarily vertical) intersects both cones to form two unbounded curves (each called a branch of the h yperbola).

Degenerate Conics

There are various ways for a plane and the cones to connect, forming what Degenerate conics are defined as a point, one line, or two lines.

Circle

• A circle may also be considered a special kind of ellipse (for the special case when the tilted plane is horizontal).
• The plane intersect horizontally to the nappe of the cone, forming a special kind of ellipse where it's point is equidistant to it's radius.
• The A and C have same coefficient and sign.
• r > 0 means a normal circle.
• r = 0 represents a degenerate circle (a point).
• r < 0 means that the point lies in the opposite direction from where it would be if r were positive, with a magnitude greater than 1.

Parts of a Circle

Radius - The distance from the center of the circle to any point on its circumference.

Center - The point equidistant from all points on the edge of the circle.

Center at (h, k):

Center at Origin(0, 0):

Standard Form of a Circle

Center(0,0): x² + y²= r²

Center(h,k): ( x − h )² + ( y − k )² = r²

General Form of a Circle

Ax²+ Bxy + Cy² + Dx + Ey + F = 0 or Ax² + Cy² + Dx + Ey + F = 0

Ellipse

• An ellipse is a type of conic section formed when a plane intersects a cone at an angle that is neither parallel nor perpendicular to the cone's base. It is the set of all points in a plane where the sum of the distances from two fixed points, called foci, is constant. In an ellipse, the major axis is the longest diameter, passing through both foci, while the minor axis is perpendicular to it at the center. An ellipse appears oval-shaped, and its eccentricity, which measures the deviation from being a circle, is always between 0 and 1.

Parts of Ellipse

Center - The central point around which the shape is symmetric. It’s located at the intersection of the ellipse’s major and minor axes, and it’s usually expressed as the coordinates(h,k).

Major axis / Principal Axis - The longest diameter of ellipse that run to center and foci,where it's length is 2a.

Minor axis -The shortest diameter of ellipse that passes to the center and perpendicular to the major axis,where its length is 2b.

Vertex (V1&V2) - The two points on the principal axis, where the length of the two end points of vertex is called major axis.

Co-vertex (W1&W2) - It is the end points of minor axis that is equidistant from the center.The coordinates of co-vertex can be denoted as C = ( 0, ±b).

Foci (F1&F2) - It is the two fixed point on the major axis of ellipse that is equidistant from the center. The coordinates of foci can be denoted as (c, 0) and (-c, 0).

Center at (h, k):

Horizontal

Vertical

Center at (0, 0):

Horizontal

Vertical

Standard Form of Ellipse

Equation of Ellipse:

CENTER	EQUATION	GRAPH
(0, 0)	$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$	Horizontal
(0, 0)	$\frac{x^2}{b^2}+\frac{y^2}{a^2}=1$	Vertical
(h, k)	$\frac{{\mathrm{(x-h)}}^2}{a^2}+\frac{{\mathrm{(y-k)}}^2}{b^2}=1$	Horizontal
(h, k)	$\frac{{\mathrm{(x-h)}}^2}{b^2}+\frac{{\mathrm{(y-k)}}^2}{a^2}=1$	Vertical

Relation of A and C

A and C have the same sign and A ≠ C.

Standard Form of Ellipse

Center (0, 0): $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$

Center (h,k): $\frac{{\left(x-h\right)}^2}{a^2}+\frac{{\left(y-k\right)}^2}{b^2}=1$

General Form of Ellipse:

Ax² + Bxy + Cy² + Dx + Ey + F = 0 or Ax² + Cy² + Dx + Ey + F = 0

Solving a, a², 2a, b, b², 2b, c, c², 2c

Step 1: Identify and calculate the axis.

major axis(a)

minor axis(b)

Step 2: Calculate a² and b².

a² = √a²

b² = √b²

Step 3: Focal distance(c).

c² = a² - b²

Solve for c

√c² = √(a²-b²)

Step 4: 2a, 2b, 2c

2a = 2 × a, where 2a is the length of the major axis.

2b = 2 × b, where 2b is the length of the minor axis.

2c = 2 × c, where 2c is the the length or the distance of the foci.

Parabola

• A parabola is a type of conic section created when a plane cuts through a cone parallel to its side. It consists of all the points in a plane that are equidistant from a fixed point, called the focus, and a fixed line, known as the directrix. This unique property gives the parabola its characteristic U-shape. Depending on how the cone is positioned, a parabola can open upwards, downwards, or sideways.

Parts of Parabola

Vertex - the sharpest turn point of the parabola.

Focus - a point that determines or defines a parabola. The distance from the focus to the vertex is always specified by p.

Latus Rectum - a line that passes through the focus and is perpendicular to the symmetry axis, with two end points. 4p always determines the distance between the latus rectum endpoints.

Axis of Symmetry - a line that cuts the parabola in half.

Directrix - a perpendicular line to the symmetry axis.

Center at (h, k):

Center at (0, 0):

Orientation of Parabola

• Upward: If the leading coefficient is more than zero, the parabola opens upwards.
• Downward: If the leading coefficient is less than zero, the parabola will open downward.
• Right: The equation illustrates a parabola with a single vertex that opens to the right if the constant is positive.
• Left: The equation illustrates a parabola with a single vertex that opens to the left if the constant is negative.

Standard Form of Parabola

Center at the origin:

VERTEX	EQUATION	OPENING
(0, 0)	x² = 4py	Upward
(0, 0)	x² = -4py	Downward
(0, 0)	y² = 4px	Right
(0, 0)	y² = -4px	Left

Center at (h, k)

VERTEX	EQUATION	OPENING
(h, k)	(x-h)² = 4p(y-k)	Upward
(h, k)	(x-h)² = -4p(y-k)	Downward
(h, k)	(y-k)² = 4p(x-h)	Right
(h, k)	(y-k)² = -4p(x-h)	Left

Understanding p and 4p

1. p: The distance from the vertex to the focus or from the vertex to the directrix of the parabola.

If p > 0, the parabola opens upward (if vertical) or to the right (if horizontal).

If p < 0, the parabola opens downward (if vertical) or to the left (if horizontal).

2. 4p: This is a constant used in the standard form of the equation of the parabola. Specifically, it is the coefficient that relates to the focus and directrix.

How to Solve for 4p or p

To solve for p, you need the equation of the parabola in standard form:

Step 1: Identify the form of the parabola.

For a vertical parabola: (x − h)² = 4p(y − k)

For a horizontal parabola: (y − k)² = 4p(x − h)

Step 2: Solve for p.

Compare the equation of the parabola with the standard form.

Identify 4p from the equation and divide by 4 to find p.

Hyperbola

• A hyperbola is a type of conic section formed when a plane intersects a double cone at an angle that is steeper than the angle of the cone’s sides. This creates two separate curves, called branches, that are mirror images of each other. Each point on a hyperbola has a unique relationship to two fixed points, known as the foci, such that the difference in distances from any point on the hyperbola to the two foci is constant.

Parts of Hyperbola

Center - The midpoint between two foci.

Vertices (V1&V2) - The points where the hyperbola intersects its transverse axis.

Conjugate axis - a line that passes through the center of the hyperbola and perpendicular to the transverse axis.The length of conjugate axis js 2b.

Endpoints of conjugate axis (W1&W2) - Points that is located at conjugate axis and perpendicular to transverse axis.

Foci (F1&F2) - Two fixed points along the transverse axis which is used to identify the focus of ellipse.

Asymtotes - Lines that passes through the center that the hyperbola approaches but never intersects.

Transverse axis- A line segment that is connected between two vertices of hyperbola with center (h,k).

Center At (h, k):

Horizontal

Vertical

Center at (0, 0):

Horizontal

Vertical

Standard Form of Hyperbola

Relation of A and C

A and C have opposite sign.

Center at the origin(0, 0):

PARTS	$\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$	$\frac{y^2}{a^2}-\frac{x^2}{b^2}=1$
Vertices	(a, 0), (-a, 0)	(0, a), (0, -a)
Endpoints of Conjugate Axis	(0, b), (0, -b)	(b, 0), (-b, 0)
Foci	(c, 0), (-c, 0)	(0, c), (0, -c)
Equation of Asymptotes	y = ±(b/a) * x	y = ±(a/b) * x
Orientation	Horizontal	Vertical

Center at (h, k):

PARTS	$\frac{{\mathrm{(x-h)}}^2}{a^2}-\frac{{\mathrm{(y-k)}}^2}{b^2}=1$	$\frac{{\mathrm{(y-k)}}^2}{a^2}-\frac{{\mathrm{(x-h)}}^2}{b^2}=1$
Vertices	(h ±a, k)	(h, k ±a)
Endpoints of Conjugate Axis	(h, k+b), (h, k-b)	(h+b, k), (h-b, k)
Foci	(h ±c, k)	(h, k ±c)
Equation of Asymptotes	y = k ±(b/a) (x-h)	y = k ±(a/b) (x-h)
Orientation	Horizontal	Vertical