ENGG MATH 1 – Precalculus

ANALYTIC GEOMETRY GENERAL FORM OF THE EQUATION OF A CIRCLE

LESSON TEN:
ANALYTIC GEOMETRY
A. The Circle
A. THE CIRCLE - The equation of a circle is always of the second degree.
a) Standard Form - The most general form of the second degree in x and y may be written in this form:
b) General Equation A circle is defined as the locus of a point P moving so that -
B. Conic Sections its distance from a fixed point is constant. The fixed point is
a. Parabola
𝑨𝒙𝟐 + 𝑩𝒙𝒚 + 𝑪𝒚𝟐 + 𝑫𝒙 + 𝑬𝒚 + 𝑭 = 𝟎
b. Hyperbola called the center and the constant distance, the radius.
c. Ellipse For a circle, it is convenient to take as the general form, the equation

OBJECTIVES: 𝒙𝟐 + 𝒚𝟐 + 𝑫𝒙 + 𝑬𝒚 + 𝑭 = 𝟎
At the end of the lesson, the
student should be able to:
TLO 10: Analyze and solve
problems involving Circle Theorem: An equation of the second degree in which x2 and y2 have equal coefficients and the
and the conic sections. xy-term is missing represents a circle (exceptionally, a single point, or no locus).
References:
- Analytic Geometry, Agalabia
and Ymas Example.
- Analytic Geometry, Gordon 1. Find the center and radius of the circle
Fuller 𝟒𝒙𝟐 + 𝟒𝒚𝟐 − 𝟒𝒙 + 𝟐𝒚 + 𝟏 = 𝟎
- Analytic Geometry, Love and
Rainville Given a circle of radius a with center at C(h,k), assume a
- Introduction to Analytic point P(x,y) on the curve then 2. Find the equation of the circle passing through the three points (0, 2), (3, 3), and(-1,1).
Geometry and Calculus
Deauna and Lamayo.1999.
Sibs Publishing House, Inc. 𝑪𝑷 = √(𝒙 − 𝒉)𝟐 + (𝒚 − 𝒌)𝟐 3. Write the equation of the circle
Philippines a. with center at (2,-3), radius 5.
If the center is at the origin,
Reminders: b. With center (2a, a) and touching the y-axis.
FINALS QUIZ NO. 2
After this discussion 𝒂 = √𝒙𝟐 + 𝒚𝟐
c. With center at (-1,-3) and passing through (-2,0)

d. With the points (2,5), (6,-1) as ends of a diameter

Hence, the equation of a circle of radius a is:

4. Draw the circles
• If the center is the origin,
a. 𝒙𝟐 + 𝒚𝟐 − 𝟒𝒙 − 𝟔𝒚 = 𝟏𝟐
𝒙𝟐 + 𝒚 𝟐 = 𝒂 𝟐
b. 𝒙𝟐 + 𝒚𝟐 = 𝟔𝒙 − 𝟖𝒚

• If the center is the point (h,k)

CONIC SECTIONS
The path of a point which moves so that its distance from a fixed point is in constant ratio to its
(𝒙 − 𝒉)𝟐 + (𝒚 − 𝒌)𝟐 = 𝒂𝟐
distance from a fixed line is called a conic section, or simply a conic.

Example.
The fixed point is called the focus of the conic, the fixed line the directrix, and the constant ratio
1. Find the center and the radius of the circle with the following equation
the eccentricity. The line through the focus parallel to the directrix intersects the curve in two
a. 𝒙𝟐 + 𝒚𝟐 − 𝟐𝒙 + 𝟒𝒚 − 𝟑 = 𝟎
points: the chord Q1Q2 joining these points is called the latus rectum, or right chord.
b. 𝒙𝟐 + 𝒚𝟐 + 𝟔𝒙 − 𝟖𝒚 − 𝟓 = 𝟎

2. Find the equation of the circle with center at (2, 1), and radius equal to 4.

Prepared by: Engr. Caroline Bautista-Moncada ENGGMATH 1

 STANDARD FORMS: Parabola with vertex (h,k)

If the axis is parallel to Ox and the curve opens to

the right

(𝒚 − 𝒌)𝟐 = 𝟒𝒂 (𝒙 − 𝒉)

If the axis is parallel to Ox and the curve opens to

the left
(𝒚 − 𝒌)𝟐 = −𝟒𝒂 (𝒙 − 𝒉)
The conic sections fall into three classes as follows:
If e<1, the conic is an ellipse;
If the axis is parallel to Oy and the curve opens to
e=1, the conic is a parabola;
the upward
e>1, the conic is a hyperbola.
(𝒙 − 𝒉)𝟐 = 𝟒𝒂 (𝒚 − 𝒌)

If the axis is parallel to Oy and the curve opens to

THE PARABOLA
the upward
(𝒙 − 𝒉)𝟐 = −𝟒𝒂 (𝒚 − 𝒌)
A parabola is defined as the locus of a point P moving so that its distance from the focus equals
its distance from the directrix.
Exercises:
1. Determine at sight the direction in which the curve opens, locate the vertex, focus, and
ends of the latus rectum and draw the curve.
a. y2=7x
b. x2+8y=0
c. (y-3)2=12(x+2)
2. Find the equation of the parabola
a. With vertex at O, axis Ox, and passing through (3,-6).
b. With vertex (-2,3) and focus (-4,3).
c. With vertex (2,-3) and directrix y= -7.

The parabola is the curve formed from all the points (x, y) that are equidistant from REDUCTION TO STANDARD FORM
the directrix and the focus. The line perpendicular to the directrix and passing through the focus Theorem: An equation of the second degree in which the xy-term is missing and only one square
(that is, the line that splits the parabola up the middle) is called the "axis of symmetry". The term is present represents a parabola with its axis parallel to a coordinate axis.
point on this axis which is exactly midway between the focus and the directrix is the "vertex"; the 𝑪𝒚𝟐 + 𝑫𝒙 + 𝑬𝒚 + 𝑭 = 𝟎 ,𝑪 ≠ 𝟎
vertex is the point where the parabola changes direction.
𝑨𝒙𝟐 + 𝑫𝒙 + 𝑬𝒚 + 𝑭 = 𝟎 ,𝑨 ≠ 𝟎
The name "parabola" is derived from a New Latin term that means something similar to "compare" Example.
or "balance", and refers to the fact that the distance from the parabola to the focus is always 1. Reduce the equation to standard form, plot the vertex, focus and ends of the latus rectum,
equal to (that is, is always in balance with) the distance from the parabola to the directrix. trace the curve.
a. 𝑦 2 − 12𝑥 + 24 = 0
The vertex is exactly midway between the directrix and the focus. b. 𝑥 2 − 2𝑥 + 2𝑦 + 7 = 0
c. 2𝑥 2 − 2𝑥 + 𝑦 − 1 = 0
2. Find the equation of the parabola with axis parallel to Ox and passing through (5,4), (11,-2),
(21,-4).

Prepared by: Engr. Caroline Bautista-Moncada ENGGMATH 1

 a = semi-major axis (distance of C to V1 or V2)
THE ELLIPSE b= semi-minor axis (distance of C to B1 or B2)
c= distance of F1 or F2 from C (h,k)
An ellipse is the locus of a point which moves so that the sum of its distances (length of the major
axis) from two fixed points (foci) is constant. It is a conic section for which e<1. 𝒄 = √𝒂𝟐 − 𝒃𝟐

GENERAL FORM
𝑨𝒙𝟐 + 𝑪𝒚𝟐 + 𝑫𝒙 + 𝑬𝒚 + 𝑭 = 𝟎 where A and B have the same algebraic sign d= distance of L to c

𝒂 𝒄 𝒂𝟐
𝒅= 𝑏𝑢𝑡 𝒆 = 𝑡ℎ𝑒𝑟𝑒𝑓𝑜𝑟𝑒 𝒅=
STANDARD FORM (Referred to the axis) 𝒆 𝒂 𝒄
(𝒙 − 𝒉)𝟐 (𝒚 − 𝒌)𝟐
• (a>b) major axis parallel to Ox + =𝟏 𝟐𝒃𝟐
𝒂𝟐 𝒃𝟐 Length of the Latus Rectum:
𝒂
• (a>b) major axis parallel to Oy
(𝒚 − 𝒌)𝟐 (𝒙 − 𝒉)𝟐
+ =𝟏 Distance of the ELR from the focus 𝒃𝟐
𝒂𝟐 𝒃𝟐
PARTS OF AN ELLIPSE 𝒂

Exercises: Draw the complete ellipse

(𝑥−1)2 𝑦2
a. + =1
25 16
𝑥2 𝑦2
b. + =1
144 169

FINDING THE EQUATION OF AN ELLIPSE:

1. Determine the major axis (using the standard forms as arbitrary equations)
2. Determine a, b, h, and k then substitute in the arbitrary equation
3. Simplify the equation to its general form.

Exercises: Find the equations of the following ellipses:

1. With eccentricity, e=1/3 and distance between foci 2.

2. Major axis 8, distance between foci 6.
3. Distance between foci 2, between directrices 8.
4. Latus Rectum 4, distance between foci 4√2.
Center: C(h,k) 5. With center at (3,2) , vertex at (8,2) and focus at (-1,2).
Foci: F1, F2 6. With vertices at (2,8) and (2,0), major axis parallel to y-axis.
Vertices:
V1 & V2 : ends of the major axis
B1 & B2 : ends of the minor axis
ELRS: Ends of the Latus Recta
Directrices: L1 & L2

Prepared by: Engr. Caroline Bautista-Moncada ENGGMATH 1