Conic Sections PDF

CONIC SECTIONS ARISTOTLE | DALTON | EINSTEIN | RUTHERFORD  CONIC SECTIONS Two millennia ago, Apollonius of Perga, the great Greek geometer, studied the curves formed by the intersection of a plane and a double right circular cone, and discovered many properties of these curves. These curves wer...

CONIC SECTIONS ARISTOTLE | DALTON | EINSTEIN | RUTHERFORD  CONIC SECTIONS Two millennia ago, Apollonius of Perga, the great Greek geometer, studied the curves formed by the intersection of a plane and a double right circular cone, and discovered many properties of these curves. These curves were later known as conic sections because they were formed from a double right circular cone.  CONIC SECTIONS The cone was thought of as having two parts that extended infinitely in both directions. A line lying entirely on the cone is referred to as a generator of the cone. All generators of a cone pass through the intersection of the two parts called the vertex.  TYPES OF CONIC SECTIONS There are three types of conics, depending on the number of generators that are parallel to the cutting plane, as shown in the figures. 1. Parabola. 2. Ellipse / Circle 3. Hyperbola  TYPES OF CONIC SECTIONS 1. If the cutting plane is parallel to one and only one generator, the curve is parabola. 2. If the cutting plane is not parallel to any generator, the curve is an ellipse. if the cutting plane is not parallel to any generator but is perpendicular to the axis, the ellipse becomes a circle. 3. If the cutting plane is parallel to two generators, the curve is a hyperbola. TYPES OF CONIC SECTIONS DEGENERATE CONIC SECTIONS 1. Point 2. Line 3. Intersecting lines  CONIC SECTIONS Conic sections are particular class of curves that are mostly found in nature. Their applications are essential in many fields of sciences as well as engineering and architecture.  CONIC SECTIONS Parabolic designs are used in a number of state-of-the-art equipment such as spotlight, radar antenna, and satellite dishes.  CONIC It is a set of points whose distances from a fixed point are in constant ratio to their distances from a fixed line that is not passing through the fixed point.  CONIC In dealing with conic section, it is important to take note of the following important elements: i. focus (F) - the fixed point of the conic ii. directrix (d) - the fixed line d corresponding to the focus. iii. principal axis (a) - the line that passes through the focus and perpendicular to the directrix. Every conic is symmetric with respect to its principal axis. iv. vertex (V) - the point of intersection of the conic and its principal axis  CONIC v. eccentricity (e) - the constant ratio. If point P is one of the points of the conic with point Q as its projection on d, then the eccentricity is the ratio of the distance |FP| to the distance |QP| is a constant. In symbols,  TYPES OF CONIC SECTIONS The three types of conics are distinguished by the value of its eccentricity: i. The conic is a parabola if the eccentricity e = 1. ii. The conic is an ellipse if the eccentricity e < 1. iii. The conic is a hyperbola if the eccentricity e > 1.  CIRCLE It is a set of all coplanar points such that the distance from a fixed point is constant. The fixed point is called the center of the circle and the constant distance from the center is called the radius of the circle.  CIRCLE Deriving the equation of a circle whose center is (0,0) and with the radius r, let P(x,y) be one of the points in the circle. This equation is referred as the standard form of equation of a circle. The equation od the circle whose center is at the point (h,k) and with the radius r  CIRCLE Determine the standard form of the equation of the circle given its center and the radius. a. C(0,0) , r : 5 b. C (-2,7) , r : 4 c. C (-8, -5) , r :3  CIRCLE The standard form of equation of a circle can be presented in another form. By squaring the binomials: where This equation is called the general form of the equation of the circle.  CIRCLE Solving for r in terms of D, E, and F in the general form:  CIRCLE Solving for r in terms of D, E, and F in the general form: But implies that and implies that CIRCLE then graph of the equation is a circle. then graph of the equation is a point circle. the equation has no graph.  SEATWORK I. Determine the standard form of equation of the circle given its center and radius. Draw its graph. 1. Center (3, -2) , r = 4 2. Center (0, 8) , r = 5 II. Write the equation of a circle in general form given its center and radius. Then, draw its graph. 1. Center (-2, -4) , r = 4 2. Center (1, -2) , r = 5/2  SEATWORK III. Determine the center and radius of each circle in general form. Then draw its graph.  SEATWORK IV. Determine whether each equation represents a circle, a point circle, or has no graph.  SEATWORK IV. Determine whether each equation represents a circle, a point circle, or has no graph.  SEATWORK V. Find the equation of the circle described in each of the following:  sample I. Determine the standard form of equation of the circle given its center and radius. Draw its graph. 1. Center (-12, -7) , r = 11 II. Write the equation of a circle in general form given its center and radius. Then, draw its graph. 2. Center (-2, -4) , r = 4 III. Determine the center and radius of each circle in general form. Then draw its graph.