Precalculus: CONIC SECTIONS: CIRCLES Von Christopher G. Chua Grading System: Attendance 10%, Class Discussions 10%, Written Works 30%, Major Exam 50% In case of absences, you are required to submit a discussion and reflection paper on the topics discussed during the absence.

Important Course Concerns Contact me directly through email: von_christopher_chua@dlsu.edu. ph vaughnchua@gmail.com All learning resources, additional readings, assignment and paper details, and important announcements will be relayed through: MATHbyCHUA: mathbychua.weebly.com This slideshow presentation will be made available through the class’s official website, mathbychua.weebly.com. The site will also provide access to download this file in printable format.

Session Objectives For this two-hour period, SHS students in Precalculus are expected to develop the following learning competencies: illustrate the different types of conic sections: parabola, ellipse, circle, hyperbola, and degenerate cases; define a circle; determine the standard form and general form of the equation of a circle; and graph a circle in a rectangular coordinate system. This slideshow presentation will be made available through the class’s official website, mathbychua.weebly.com. The site will also provide access to download this file in printable format.

Intersections of cones and planes Conic Sections 1 Intersections of cones and planes

Starting with Basics Explain the following mathematical terms: Undefined Terms in Geometry Intersecting lines Parallel lines Perpendicular lines Angle Cone The three undefined terms in Geometry are the point, line, and plane. A point is represented by a dot, a line is a set of points that form a straight line, and a plane is a perfectly flat surface extending infinitely in every direction. Intersecting lines are two or more lines that meet at exactly one point. Parallel lines are lines that do not intersect, even if extended infinitely in both directions. Perpendicular lines are special type of intersecting lines that intersect to form a right angle. An angle is composed of two rays with a common endpoint. A cone is a geometric solid composed of a point called the vertex and a circular base.

What are CONIC SECTIONS? If a plane intersects a double right circular cones, the intersection is a two-dimensional curve of different types. These curves are called conic sections. Also called conics, these different types are: PARABOLA CIRCLE ELLIPSE HYPERBOLA

What are CONIC SECTIONS? PARABOLA CIRCLE ELLIPSE HYPERBOLA Circle - when the plane is horizontal Ellipse - when the (tilted) plane intersects only one cone to form a bounded curve Parabola - when the plane intersects only one cone to form an unbounded curve Hyperbola - when the plane (not necessarily vertical) intersects both cones to form two unbounded curves (each called a branch of the hyperbola)

DEGENERATE CASES If the plane intersects the double right circular cones, passing through the vertex, degenerate cones are formed. When perpendicular to the base, such as that of a circle, a point is formed. When the plane is tilted, such as the case of a parabola, a line is the intersection. A hyperbola becomes two intersecting lines.

A circle is the set of all points equidistant to a fixed point. center CIRCLES A circle is the set of all points equidistant to a fixed point. radius

EQUATION OF A CIRLE Consider a circle plotted on the Cartesian Coordinate plane. Determine the coordinates of the center and its radius.

EQUATION OF A CIRLE 𝑃 (𝑥, 𝑦) Assume that we do not know the actual coordinates of the center. What if the center is at the point of origin? 𝐶 (ℎ, 𝑘)

STANDARD FORM of the equation of a circle The standard form of the equation of a circle whose center is at the origin with radius, r, is 𝑥 2 + 𝑦 2 = 𝑟 2 For any circle whose center have coordinates (h, k) with radius, r, the standard form of its equation is 𝑥−ℎ 2 + 𝑦−𝑘 2 = 𝑟 2

STANDARD FORM of the equation of a circle Give the standard form of the equation of the circle described in each item below. 𝐶 (0, 0), 𝑟=8 Center at the origin, 𝑟= 5 3 𝐶 4, 1 and containing 𝑃 5, −2 𝐶 −3, −5 and containing 𝑃 1, 0

EQUATION OF A CIRLE Solve for the equation of the following circles: circle A circle B center (5,−6), tangent to the y- axis center (5,−6), tangent to the x- axis has a diameter with endpoints A(−1, 4) and B(4, 2)

IN, ON, or OUT Consider the circle with equation, 𝑥+2 2 + 𝑦−4 2 =100 𝑥+2 2 + 𝑦−4 2 =100 Determine the location of the following points relative to the circle. P (-2, 14) Q (-6, 6) R (-10, 9) S (7, 9)

GENERAL FORM of the equation of a circle The general form of the equation of a circle is 𝐴𝑥 2 + 𝐴𝑦 2 +𝐶𝑥+𝐷𝑦+𝐸=0 where 𝐴≠0

GENERAL FORM of the equation of a circle Transform the following equations into the general form: 𝑥−2 2 + 𝑦+3 2 =11 𝑥+15 2 + 𝑦−1 2 =1 𝑥−2 2 + 𝑦 2 =7 𝑥+9 2 + 𝑦−4 2 =13 𝑥− 3 4 2 + 𝑦+ 5 4 2 =2

Transforming equations Transform the following equations into the standard form: 𝑥 2 + 𝑦 2 −6𝑥−7=0 𝑥 2 + 𝑦 2 −14𝑥+2𝑦=−14 16 𝑥 2 +16 𝑦 2 +96𝑥−40𝑦−315=0

Transforming equations 𝑥 2 + 𝑦 2 +6𝑥+16𝑦+48=0 𝑥 2 +6𝑥 + 𝑦 2 +16𝑦 =−48 1 2 6 =3, 3 2 =9 1 2 16 =8, 8 2 =64 𝑥 2 +6𝑥+9 + 𝑦 2 +16𝑦+64 =−48+9+64 𝒙+𝟑 𝟐 + 𝒚+𝟖 𝟐 =𝟐𝟓 Group the terms based on the variables. Complete the square to make each group of terms a perfect square trinomial (PST). Add the constants of the PSTs to the right side. Express the PSTs as squares of binomials