PRE CALCULUS

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CONIC SECTIONS AND CIRCLES

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DEGENERATE CONIC SECTIONS

 

HOW TO IDENTIFY THE FOUR CONIC SECTIONS IN EQUATION FORM

  • Circle. When x and y are both squared and the coefficients on them are the same — including the sign.

    For example, take a look at 3x2 – 12x + 3y2 = 2. Notice that the x2 and y2 have the same coefficient (positive 3). That info is all you need to recognize that you’re working with a circle.

  • Parabola. When either x or y is squared — not both.

    The equations y = x2 – 4 and x = 2y2 – 3y + 10 are both parabolas. In the first equation, you see an x2 but no y2, and in the second equation, you see a y2but no x2. Nothing else matters — signs and coefficients change the physical appearance of the parabola (which way it opens or how fat it is) but don’t change the fact that it’s a parabola.

  • Ellipse. When x and y are both squared and the coefficients are positive but different.

    The equation 3x2 – 9x + 2y2 + 10y – 6 = 0 is one example of an ellipse. The coefficients of x2 and y2 are different, but both are positive.

  • Hyperbola. When x and y are both squared, and exactly one of the coefficients is negative and exactly one of the coefficients is positive.

    The equation 4y2 – 10y – 3x2 = 12 is an example of a hyperbola. This time, the coefficients of x2 and y2 are different, but exactly one of them is negative and one is positive, which is a requirement for the equation to be the graph of a hyperbola.

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CIRCLE

Formula of a Circle

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STANDARD FORM TO GENERAL FORM

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GENERAL FORM TO STANDARD FORM

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EQUATION OF A CIRCLE THROUGH 3 POINTS

Example :

Find the center point and radius for the equation of a circle passing through (2,2), (2,4) and (5,5) x and y-coordinates.

Given :

A(2,2) B(2,4) C(5,5)

Solution :
Step :1

Substitute the given x and y-coordinates in the circle formula,

(2 – h)2 + (2 – k)2 = r2 ———— (1)

(2 – h)2 + (4 – k)2 = r2 ———— (2)

(5 – h)2 + (5 – k)2 = r2 ———— (3)

Step :2

Let us find the value of k by simplifying the first (1) and second (2) equation, (2 – h)2 + (2 – k)2 = (2 – h)2 + (4 – k)2 4 – 4h + h2+ 4 – 4k + k2 = 4 – 4h + h2+16 – 8k + k2 8 – 4k = 20 – 8k k=3

Step :3

Now, let us find out the value of h by simplifying the second (2) and third (3) equation (2 – h)2 + (2 – k)2 = (5 – h)2 + (5 – k)2 4 – 4h + h2+ 4 – 4k + k2 = 25 – 10h + h2+ 25 – 10k + k2 8 – 4k – 4h = 50 – 10h – 10k 6k + 6h = 42 Substitute k=3 in equation 6h = 24 h=4 Therefore center point is c(h,k) = c(4,3)

Step :4

Substitute h,k values in the given formula r2 = (x – h)2 + (y – k)2 r2 = (2 – 4)2 + (2 – 3)2 r2 = (-2)2 + (-1)2 r2 = 5 r = 2.24

Step :5

Substitute h, k values in the circle formula circle equation = (x – h)2 + (y – k)2 circle equation = (x – 4)2 + (y – 3)2

Result :

Center point is c(h,k) = c(4,3) Radius of a Circle r = 2.24 Circle Equation = (x – 4)2 + (y – 3)2 = (2.24)2

 

TANGENT TO A CIRCLE

A tangent to a circle is a straight line which touches the circle at only one point. This point is called the point of tangency.

The tangent to a circle is perpendicular to the radius at the point of tangency.

 

PARABOLA

CONCEPT 1 – The Definition of a Parabola

 

As mentioned in the previous objective the conic sections represent the curves that can be found from the cross sections of the cone or double cone – the circle, parabola, ellipse, and hyperbola.  A parabolic curve can be found by cutting into the cone so that it passes through the lateral face and the base.

 

While most students could tell you the basic definition of a circle before entering geometry, very few could tell you the definition of a parabola.

A parabola is the locus (set) of points that are equidistant from a fixed point,
called the focus, and a straight line, called the directrix.
The definition of a parabola even after hearing it is still quite illusive to students because it is very hard to visualize why this would create that particular curve.  Breaking down the definition geometrically is very important to helping them understand it.
It is important to remember that the distance from a fixed point to a line is found in the perpendicular distance.  The shortest distance from any point to a line is the perpendicular distance.
In an attempt to visualize the points that satisfy this definition most students see the midpoint of the perpendicular distance from the focal point to the directrix.  This is only one point of the parabola but actually it is a very important one.  It is the vertex of the parabola.  The vertex will either be the minimum or maximum value of the parabola depending on its orientation in the plane.

 

 

CONCEPT 2 – The Construction of a Parabola
Before going to the coordinate plane to derive an equation for the parabola, I think it is important to first be able to construct it.  The physical construction of the parabola will help us to understand how to approach the equation on the coordinate plane.
I want to emphasize two important concepts in the definition.
(1) For a point to be equidistant to two points it must be on the perpendicular bisector of those two points.
(2) If one of the distances is a distance to a line, then it will be a perpendicular distance to that line.
You will see both of these concepts played out in the construction.

 

 

CONCEPT 3 – The Equation of a Parabola

To arrive at the equation for the parabola we use its definition.  To set this up we generate coordinates that represent these relationships.  For simplicity we will use the origin (0,0) as the vertex.  This creates a focal point that is directly above it, A(0,p) and a directrix that is the same distance from the vertex but in the opposite direction, y = -p.  Traditionally the distance to the focal point from the vertex is labelled as p.

A couple of things to notice in our derivation of this equation:
(1) We arranged it so that the vertex of the parabola would be at the origin (0,0).  It is the exact midpoint of the perpendicular distance from the focal point A (0,p) and the directrix, y = -p.  In this objective we will only work with parabolas with a vertex at the origin.
(2) The value of p represents the distance from the vertex to the focal point and the distance from the vertex to the directrix in the opposite direction.

 

 

CONCEPT 4 – Understanding the Equation of a Parabola
As with all equations the x and y values represent the x and y coordinates of the points that lie on this curve.  This leaves the p value as the only thing to discuss.  The value of p, the distance from the vertex to the focal point, greatly alters the shape of the parabola.  We just learned that the positive or negative orientation of p determines the direction that it opens.  Below we will see that the size of p will also alter the parabola.

 

 

Understanding the forms of the equation and how the focal point, vertex and directrix relate to each other allows us to determine information from each other.  The following examples display this connectivity.

 

 

The vertex and the p value allow us to determine anything we want to know about the parabola.
Working backwards is often more of a challenge for students, when given the equation and asked to determine the p value and the other key points of the parabola.  You would think that this would be easier but it is not because the equation can be written in different forms and the solving often involves multiplying and dividing fractions (oh no…. not fractions!!!).  The following examples display this process.

 

 

 

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Published by Reuben Pangilinan

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