Conic Sections

Table of Contents Sections of a right cone The Conics in everyday life Terminology Conics as plane loci - Problems Double Hyperbola Conics as plane loci Four Conic Sections Conics in a rectangle More Problems Tangent to the Conics 1 Focal Spheres Tangents to the Conics 2 (HL) Common Properties Centre of Curvature & Evolute

FOUR CONIC SECTIONS

The circle, ellipse, parabola and hyperbola are known as conic sections All four curves are obtained by slicing a double cone at varying angles

When a plane is parallel to one generator, the section is a parabola When a plane cuts all generators on one side of the apex, the section is an ellipse When a plane cuts both parts of the double cone, the section is a hyperbola When a plane is parallel to one generator, the section is a parabola When this plane is perpendicular to the axis, the section is a circle CONICS AS SECTIONS OF A CONE Ellipse Circle Parabola Hyperbola The parabola has only one branch and is unlimited The hyperbola has two branches extending indefinitely in opposite directions The ellipse and circle are closed curves 5

A circle is obtained when the cutting plane is perpendicular to the axis of the cone

An ellipse is obtained when the cutting plane cuts all elements of the upper or lower cone

A parabola is obtained when the cutting plane is parallel to a generating line

A hyperbola is obtained when the cutting plane cuts both the upper and lower cones

THE ELLIPSE AROUND US The circle is the simplest of all curves However, the ellipse is the curve most often "seen" in everyday life This is because every circle, viewed obliquely, appears elliptical The orbits of the planets are ellipses

THE PARABOLA AROUND US When a sliothar or golf ball is hit into the air, it follows a parabolic path The centre of gravity of a leaping salmon describes a parabola The parabola is used by engineers in designing some suspension bridges

THE HYPERBOLA AROUND US A hyperbola is formed when a right circular cone is intersected by a plane parallel to its axis Such an intersection occurs in physical situations as simple as sharpening a pencil that has a polygonal cross section………. …...and in the patterns formed on a floor by a flash light

SECTIONS OF A RIGHT CONE

PARABOLA vertex parabola cutting plane parallel to element of cone true shape of cut surface horizontal cutting plane yields a circle in plan double ordinate

ELLIPSE centre line true shape of cut surface is an ellipse cutting plane passing through all elements of the cone

ELLIPSE File - 24.swf

True Shape Of Cut Surface Datum Line

cutting plane parallel to axis of cone HYPERBOLA cutting plane parallel to axis of cone true shape of cut surface is a hyperbola 45°

FOCAL SPHERES

TERMINOLOGY A focal sphere is a sphere inscribed in the cone tangential to the cutting plane The focus is the point of contact between the sphere and the cutting plane The directrix is the line of intersection between the cutting plane and the horizontal plane through the tangent circle of the sphere and cone The axis is a line through the focus perpendicular to the directrix The vertex is the point in which the curve intersects the axis

FOCAL SPHERE FOR A PARABOLA File Parabola.EPRT

FOCAL SPHERE FOR A PARABOLA directrix vertex Cutting plane S-S parallel to extreme generator of cone focus S vertex of cut surface axis plane which contains circle of contact between cone and sphere S focal sphere touches cutting plane at focal point

FOCAL SPHERES FOR AN ELLIPSE File - Ellipse.EPRT

FOCAL SPHERES FOR AN ELLIPSE directrices vertices cutting plane S-S inclined to extreme generator of cone focal points axis S planes which contain circle of contact between cone and sphere focal spheres touch cutting plane at focal points vertices of cut surface S

FOCAL SPHERE FOR A HYPERBOLA File - Hyperbola.EPRT

parallel to axis of cone FOCAL SPHERE FOR A HYPERBOLA plane which contains circle of contact between cone and sphere S directrix vertex of cut surface vertex focus cutting plane S-S parallel to axis of cone axis focal sphere touches cutting plane at focal point S

THE CONICS IN EVERYDAY LIFE

THE ELLIPSE AROUND US In an ellipse, any signal (light or sound) that starts at one focus is reflected to the other focus This principle is used in lithotripsy, a medical procedure for treating kidney stones The patient is placed in a elliptical tank of water, with the kidney stone at one focus High-energy shock waves generated at the other focus are concentrated on the stone, pulverizing it The electrons of an atom move in an approximately elliptical orbit with the nucleus at one focus

THE PARABOLA AROUND US If a light is placed at the focus of a parabolic mirror (a curved surface formed by rotating a parabola about its axis), the light will be reflected in rays parallel to the axis This property is used in the design of flashlights and headlights The bulb is placed at the focus for the high beam and a little above the focus for the low beam The opposite principle is used in the giant mirrors in reflecting telescopes used to collect light and radio waves from outer space The beam comes toward the parabolic surface and is brought into focus at the focal point

THE HYPERBOLA AROUND US The path of a comet often takes the shape of a hyperbola Many telescopes use hyperbolic (hyperbola-shaped) lenses Sound waves travel in hyperbolic paths, and so there are applications of the hyperbola in navigation (GPS systems) The hyperbolic shape of water cooling towers helps the tower do its job more efficiently

TERMINOLOGY

A conic section may be defined as the locus of points in a plane whose distances from the focus are in constant ratio to their distances from the directrix. This constant is called the eccentricity. axis directrix F P distance from focus F distance from directrix Eccentricity = PF PD = An ellipse has eccentricity between 0 and 1 A parabola has eccentricity 1 A hyperbola has eccentricity greater than 1

TERMINOLOGY COMMON TO ALL CONICS Ellipse Eccentricity <1 Directrix F1 F2 Normal Tangent Chord Minor Axis Focal Chord Vertex Major Axis Vertex Ordinate Minor Auxiliary Circle Major Auxiliary Circle

The normal is perpendicular to the tangent TERMINOLOGY COMMON TO ALL CONICS Eccentricity line P The normal is perpendicular to the tangent tangent normal F axis Latus Rectum Double Ordinate Parabola Eccentricity = 1 directrix

TERMINOLOGY COMMON TO ALL CONICS Double Ordinate Latus Rectum Transverse Axis Axis Vertices Focal Points Branches Hyperbola Eccentricity >1

ELLIPSE CONICS AS PLANE LOCI

The directrix and focal point of a parabola are shown. Locate the vertex and draw a portion of the curve axis F directrix

The eccentricity line is inclined at 45° to the axis PARABOLA Eccentricity = 1 The eccentricity line is inclined at 45° to the axis The endpoints of the latus rectum give two points on the curve All arcs are swung from F 45° axis vertex F directrix

The directrix and focal point of a hyperbola are shown The directrix and focal point of a hyperbola are shown. The eccentricity of the curve is Locate the vertex and draw a portion of the curve. /4. 5 axis F directrix

The endpoints of the latus rectum give two points on the curve eccentricity line 5 units F directrix axis HYPERBOLA Eccentricity > 1 The eccentricity line is inclined at an angle greater than 45° to the axis The endpoints of the latus rectum give two points on the curve All arcs are swung from F 45° vertex 4 units

The directrix and focal point of an ellipse are shown The directrix and focal point of an ellipse are shown. The eccentricity of the curve is Locate the vertex and draw a portion of the curve. /4. 3 axis F directrix

The endpoints of the latus rectum give two points on the curve 3 units eccentricity line F directrix The endpoints of the latus rectum give two points on the curve 45° 45° axis VV = major axis V V 4 units ELLIPSE Eccentricity < 1 The eccentricity line is inclined at an angle less than 45° to the axis All arcs are swung from F

The eccentricity line passes through the top of the major circle directrix ALTERNATE SOLUTION 3 units eccentricity line The eccentricity line passes through the top of the major circle major circle minor circle The endpoints of the latus rectum give two points on the curve 45° 45° axis VV = major axis V V 4 units ELLIPSE Eccentricity < 1 The eccentricity line is inclined at an angle less than 45° to the axis

CONICS AS PLANE LOCI - PROBLEMS

When a plane goes faster than the speed of sound, a sonic boom shock wave occurs. A cone shaped wave shoots out of the back of the plane as shown in the photograph over and intersects the ground in part of a hyperbola (below). It hits every point on this curve at the same time, so that people in different places along the curve on the ground hear it at the same time. D V axis In the drawing above the line DD represents the directrix of a hyperbola and V is the vertex of the curve. The eccentricity is 1.25. Locate the focal point of the hyperbola and draw a portion of the curve.

When a pencil that has a hexagonal cross section is sharpened, a hyperbola is formed. This results from the intersection of the conical point of the pencil by one of its flat sides. Two lines VF and FD are shown in the drawing below. V is the vertex of a hyperbola, F is a focal point of the curve and D is a point on the directrix. Locate the directrix and eccentricity line and draw a portion of the curve. D F V

A photograph of a Cleveland Launcher driver is shown A photograph of a Cleveland Launcher driver is shown. The underside of the club is in the shape of a parabola. In the drawing below, F is the focus of the parabola and the line AFB represents a focal chord of the curve. (a) Determine the position of the directrix and the axis of the parabola. (b) Draw the portion of the curve as shown.

ELLIPSE CONICS IN A RECTANGLE

PARABOLA vertex The same number of equal divisions must be used for each half axis 4 equal divisions 4 equal divisions

ELLIPSE 5 equal divisions minor axis major axis 5 equal divisions The same number of equal divisions must be used for each quadrant

HYPERBOLA axis V1 V2 5 equal divisions The same number of equal divisions are used 5 equal divisions

MORE PROBLEMS

A photograph of a Parabola Hall Table designed by Nathan Hunter is shown. Two curly wood panels intersect steel parabolas to support a glass top. The curved panels are portions of an ellipse. The incomplete elevation of the table is given over. The two parabolas are to be inscribed in the rectangles ABCD and EFGH, respectively. P is a point on the curve of the ellipse whose minor axis MN is drawn. Complete the view of the hall table showing all construction lines.

The St. Louis Science Centre Planetarium's exterior curved surface is in the shape of a hyperboloid. This shape is formed by revolving a hyperbola around its axis. The drawing below shows an incomplete elevation of the planetarium. The line VV represents the transverse axis of the double hyperbola and P and Q are points on the curve. Complete the elevation by drawing the double hyperbola.

TANGENTS TO THE CONICS 1

The tangent is perpendicular to the normal TANGENT TO A CIRCLE FROM A POINT ON THE CURVE An ellipse is a stretched circle…….. The tangent is perpendicular to the normal 90° P normal tangent F 1 F 2 P is joined to the foci …... and a circle is an ellipse with the two focal points located at the centre

TANGENT TO AN ELLIPSE FROM A POINT P ON THE CURVE normal F 1 F 2 The normal is perpendicular to the tangent and can be located by bisecting angle F1PF2

The normal is perpendicular to the tangent TANGENT TO A PARABOLA FROM A POINT P ON THE CURVE to other focal point at infinity P tangent normal axis F The normal is perpendicular to the tangent directrix

TANGENT TO A HYPERBOLA FROM A POINT P ON THE CURVE normal axis F 1 V 1 V 2 F 2 tangent

TANGENTS TO THE CONICS 2 (HIGHER LEVEL)

TANGENT TO A PARABOLA FROM A POINT P OUTSIDE THE CURVE axis directrix F P to other focal point at infinity point of contact tangent R Two tangents can be drawn from P to other focal point at infinity

TANGENT TO AN ELLIPSE FROM A POINT P OUTSIDE THE CURVE directrix P F 1 2 point of contact R tangent point of contact A second tangent can be drawn from P

TANGENT TO A HYPERBOLA FROM A POINT P OUTSIDE THE CURVE Radius is equal to length of transverse axis tangent point of contact V 2 1 F axis P

DOUBLE HYPERBOLA FROM FOCI AND TRANSVERSE AXIS

Draw a portion of the curve when given the focal points and a point P on the curve P V1 – V2 a b c d e f F1 V1 V2 F2 R.V2 - c R.V1 - c

Draw a portion of the curve when given the focal points and the transverse axis. R.V1 – d R.V2 - d a b c d e f F1 V1 V2 F2 R.V2 - c R.V1 - c

How to locate the asymptotes and directrices for a double hyperbola Conjugate axis Transverse axis R axis F1 V1 V2 F2 Auxiliary circle directrices asymptotes

PROPERTIES COMMON TO ALL CONICS

This property holds for all conics The angle subtended at the focus by that part of the tangent between the focus and the directrix is 90° point of contact directrix F 1 2 tangent 90°

Perpendiculars from the foci to the tangent meet the tangent on the major auxiliary circle major circle tangent axis F 1 2

V1 V2 axis F 1 2 In a double hyperbola, perpendiculars from the foci to the tangent meet the tangent on the auxiliary circle tangent auxiliary circle

In a parabola, a perpendicular from the focus to the tangent meets the tangent on the auxiliary circle axis directrix F tangent In a parabola, the major auxiliary circle is the tangent at the vertex

The line joining the midpoints of parallel chords passes through the centre of the conic two parallel chords F 1 2 centre of ellipse

The line joining the midpoints of parallel chords of a parabola is parallel to the axis directrix F to the centre of the parabola at infinity two parallel chords

CENTRE OF CURVATURE & EVOLUTE

CENTRE OF CURVATURE FOR A POINT P ON A PARABOLA axis directrix F P to second focal point normal 90° centre of curvature

CENTRE OF CURVATURE FOR A POINT P ON AN ELLIPSE 1 2 normal centre of curvature

CENTRE OF CURVATURE AT THE VERTEX OF A PARABOLA axis directrix F V C centre of curvature R circle of curvature

The distance between V and F is added CENTRE OF CURVATURE AT THE VERTEX OF AN ELLIPSE F 1 2 VF The distance between V and F is added R V centre of curvature

The distance between V and F is added CENTRE OF CURVATURE AT THE VERTEX OF A HYPERBOLA F 2 1 V The distance between V and F is added VF R centre of curvature

The evolute of a conic is the locus of its centre of curvature To construct an evolute: Determine the centres of curvatures for a number of points on the curve The points should include the vertices of the conic Plot a curve through them

EVOLUTE OF AN ELLIPSE F 1 2 P centre of curvature

The evolute of an ellipse looks like this……… 1 2

The evolute of a parabola looks like this……… axis F Find out yourself what the evolute of a hyperbola looks like!