Introduction To Conics PDF

Conic Section or Conics are curves formed by the intersection of a plane and a cone. The curves formed by this intersection depend on the angle by which a plane cut the cone. Degenerate Conics are figures formed...

Conic Section or Conics are curves formed by the intersection of a plane and a cone. The curves formed by this intersection depend on the angle by which a plane cut the cone. Degenerate Conics are figures formed when the plane cuts or intersects the vertex of the cone. Curves that do not pass through the vertex of the cone are called non-degenerate conics. Circle – the cone is cut by the plane horizontally Ellipse – the cone is cut by the plane diagonally through its side Parabola – the cone is cut by a plane diagonally from its side through its base - When a plane intersects a double cone parallel to its generator, can form parabola Hyperbola – when two cones are cut by a plane and forms two unbounded curves  Using the calculated distance as the radius, a circle is drawn around each seismic station on a map. The epicenter lies somewhere on this circle. Data from at least three seismic stations is required. The circles drawn around each station will intersect at a single point. This point of intersection represents the epicenter of the earthquake. Planets orbit the Sun in elliptical paths, with the Sun at one focus of the ellipse. The path followed by a projectile under the influence of gravity is a parabola. This principle is used in ballistics and other fields to predict the trajectory of objects in motion.  Parabolic dishes are used in satellite dishes and telescopes to focus incoming signals (radio waves or light) at a single point, the focus, allowing for efficient reception or observation. Parabolic arches are often used in bridges and architectural structures due to their strength and stability. The shape distributes weight evenly, allowing for larger spans and open spaces. Parabolic reflectors are used in headlights and flashlights to create a focused beam of light. The light source is placed at the focus of the parabola, ensuring that the reflected light rays are parallel and create a strong beam.  Hyperbolic shapes are sometimes used in the design of cooling towers at power plants. The hyperbolic shape provides structural stability and allows for efficient heat dissipation. Elliptical and hyperbolic shapes are used in the design of lenses for cameras, telescopes, and other optical instruments. The shape of the lens determines how light rays are refracted and focused, impacting the quality of the image. Questions to ponder: Why the parabolic shape is the preferred choice for satellite dishes? How environmental factors such as weather and obstructions can affect signal reception? A bridge is supported by cables that hang in the shape of a parabola. If the vertex of the parabola is at the lowest point of the bridge, where would the focus of the parabola be located?  Circle is a set of all points in a plane which are equidistant from a fixed point called center. Situational Problems Involving Circles Example 1. A street with two lanes, each 10 ft wide, goes through a semicircular tunnel with radius 12 ft. How high is the tunnel at the edge of each lane? Round off to 2 decimal places. Solution. We draw a coordinate system with origin at the middle of the highway, as shown. Because of the given radius, the tunnel's boundary is on the circle x2 + y2 = 122. Point P is the point on the arc just above the edge of a lane, so its x-coordinate is 10. We need its y-coordinate. We then solve 102 + y2 = 122 for y > 0, giving us y = 𝟐√𝟏𝟏 ≈ 𝟔. 𝟔𝟑 = 𝒇𝒕 Example 2. A piece of a broken plate was dug up in an archaeological site. It was put on top of a grid, as shown in Figure 10, with the arc of the plate passing through A(-7, 0), B(1, 4) and C(7, 2). Find its center, and the standard equation of the circle describing the boundary of the plate. Solution. We first determine the center. It is the intersection of the perpendicular Study This! A cellular tower is located at coordinates (5, 3) on a map, and it has a coverage radius of 7 kilometers. Analyze the equation of the circle representing the tower's coverage area. Determine if a cell phone user located at (10, 8) is within the tower's range. Justify your answer using the equation. According to Garces (2016), a Parabola is a symmetrical open plane curve formed by the intersection of a cone with a plane parallel to its side. We have already learned that the graph of the quadratic equation y = ax 2 + bx + c where (a ≠ 0) is a U-shaped curve called a parabola that opens either upward or downward, depending on whether the sign of a is positive or negative. To our surprise and delight, we may also define parabolas in terms of distance. As shown in Figure 2, the line from the focus (F) to a point on the curve has the same length as the line from the point on the curve to the directrix. The vertex (V) of the parabola lies halfway between the focus and the directrix, it is also the point where the parabola is mostly sharply curved. Figure 3 shows that the axis of symmetry is the line that runs through the focus perpendicular to the directrix. The latus rectum is the chord of the parabola that is parallel to the directrix and passes through the focus. Parabolas can open facing, down, left, right, or in some other arbitrary direction. Any parabola can be repositioned and rescaled to fit exactly on any other parabola—that is, all parabolas are geometrically similar. NOW IT’S YOUR TURN! Match the terms in Column A with the definitions on Column B. In all four cases below, we assume that c > 0. The vertex is V (h, k), and it lies between the focus F and the directrix l. The focus F is c units away from the vertex V , and the directrix is c units away from the vertex. Recall that, for any point on the parabola, its distance from the focus is the same as its distance from the directrix. REMEMBER THIS!