Conic Sections
Given any five points in the plane in general linear position, meaning no three collinear, there is a unique conic passing through them
Here is a Geogebra Applet for that, move the A, B, C, D, E points
Parabola
A parabola is defined as the
locus
of points that are equidistant from both the directrix (a fixed straight line)
and the focus (a fixed point)
A conic section has the eqn Ax2 + Bxy + CY2 + Dx + Ey + F = 0 where A,B,C,D,E and F are constants. For a parabola, B2 - 4AC = 0
A parabola is generally of two types
- Y-axis parabola, with equation of type x2 = 4ay
- X-axis parabola, with equation of type y2 = 4ax
(x-h)2 = 4a(y-k) is a Y-axis parabola with vertex at (h,k) and focus at (h,k+a)
Distance between focus and parabola = Distance between directrix and parabola
| Serial no. | Property | Y-axis parabola | X-axis parabola |
|---|---|---|---|
| 1. | General equation | x2 = 4ay | y2 = 4ax |
| 2. | Parametric equation | R(t) = tî + t2⁄4aĵ | R(t) = t2⁄4aî + tĵ |
| 3. | Eccentricity | e = 1 | e = 1 |
| 4. | Co-ordinates of Vertex | (h,k) | (h,k) |
| 5. | Co-ordinates of Focus | (h,k+a) | (h+a,k) |
| 6. | Equation of Principal axis | x = h | y = k |
| 7. | Equation of Tangent at point (x0, y0) | (x-h)x0 = 2a(y - k + y0) | (y-k)y0 = 2a(x - h + x0) |
| 8. | Equation of Normal at point (x0, y0) | x - x0 = -x0⁄2a (y- y0) | y - y0 = -y0⁄2a (x- x0) |
| 9. | Equation of Latus rectum | y = a | x = a |
| 10. | Equation of Directrix | y = -a | x = -a |
| 11. | Length of Latus rectum | 4a | 4a |
| 12. | Focal parameter | 2a | 2a |
Labelled parabola
A Quadratic equation, y = ax2 + bx + c when graphed also forms a
parabola.
Here are some properties of the parabola formed :-
| Serial no. | Property | Value |
|---|---|---|
| 1. | General equation | y = ax2 + bx + c |
| 2. | Co-ordinates of y intercept | (0,c) |
| 2. | Co-ordinates of Vertex | (-b⁄2a, -b2⁄4a + c ) |
| 3. | Co-ordinates of Focus | (-b⁄2a, 1 - b2⁄4a + c ) |
| 4. | Equation of Principal axis | x = -b⁄2a |
| 7. | Equation of Directrix | y = c - 1⁄4a + b2⁄4a |
Reflection
If inner surface of a parabola is polished, then light that enters it travelling parallel to the
axis of symmetry is reflected toward the focus. That is why that point is called focus
Parabolic mirrors are even better than concave mirrors in focusing light.
Thomas Edison with his parabolic Seachlight
Recording wildlife sounds using a parabolic microphone
Radio telescopes use parabloa shaped dishes
A parabola of the form y = x2 showing the reflective property
Parabolas in real life
Two soldiers loading a mortar
A jet dropping unguided bombs
The olympic flame has been traditionally lit using parabolic reflectors
Mirascope
The mirascope consists of two parabolic mirrors facing each other in a clamshell fashion. The key to the design is that the focal point of each parabolic mirror sits at the vertex of the other, and a hole is made in the top mirror's vertex where the image is produced.
Mirascope
The mirascope consists of two parabolic mirrors facing each other in a clamshell fashion. The key to the design is that the focal point of each parabolic mirror sits at the vertex of the other, and a hole is made in the top mirror's vertex where the image is produced.
Mirascope
Ellipse
An ellipse is defined as the
locus
of points whose sum of distances from two focii (fixed points) is constant.
A conic section has the eqn Ax2 + Bxy + CY2 + Dx + Ey + F = 0 where A,B,C,D,E and F are constants. For an ellipse, B2 - 4AC < 0
Sum of distances from two focii to the ellipse = constant
| Serial no. | Property | ellipse |
|---|---|---|
| 1. | General equation | x2⁄a2 + y2⁄b2 = 1 |
| 2. | Parametric equation | R(t) = a cos(θ)î + b sin(θ)ĵ |
| 3. | Co-ordinates of Center | (h,k) |
| 4. | Eccentricity | 0 < (e = c⁄a) < 1 |
| 5. | Linear Eccentricity | c = (a2 - b2)1⁄2 |
| 6. | Co-ordinates of Focii | (-c+h,k) and (c+h,k) |
| 7. | Equation of Major axis | y = k |
| 8. | Equation of Minor axis | x = h |
| 9. | Equation of Tangent at point (x0, y0) | (x0x)⁄a2 + (y0y)⁄b2 = 1 |
| 10. | Equation of Normal at point (x0, y0) | (a2x)⁄x0 - (b2y)⁄y0 = a2 - b2 |
| 11. | Equation of Directrix | x = a⁄e and x = -a⁄e |
| 12. | Length of Latus rectum | 2b2⁄a |
| 13. | Area | πab |
| 14. | Focal parameter | b2⁄c |
Labelled ellipse
Reflection
Ellipses have this unique property that if we polish the inner surface of an ellipse, then light beams passing through one of the focii will converge at the other focus.
Circles also have this property, as circles are special cases of ellipses with eccentricity = 0
Lithotripsy - A half ellipsoid shaped piece is used in this procedure. An ultrasound emitter is placed on one focus and Patient's kidney is placed on the other focus.
Whispering Galleries - These are ellipse-like shaped architectures found all around the world. The legend is that John Quincy Adams, the 6th president of U.S.A. used the acoustics of Statuary Hall, Washington DC to listen to his rival party’s conversation while he pretended to sleep. His desk was a one focal point, and the leader of the other party’s desk was at the other.
Using ultrasound to treat kidey stones
Statuary Hall, Washington DC, U.S.A.
Gol Gumbaz, Bijapur, India
An ellipse showing its reflective propery
Ellipses in real life
Accortiding to Kepler's first law, planets orbit in elliptical orbits
A rugby ball is ellipsoid in shape
An elliptical pool table
Hyperbola
A hyperbola is defined as the
locus
of points whose difference of distances from two focii (fixed points) is constant.
A conic section has the eqn Ax2 + Bxy + CY2 + Dx + Ey + F = 0 where A,B,C,D,E and F are constants. For a hyperbola, B2 - 4AC > 0
Differnce of distances from two focii to the hyperbola = constant
| Serial no. | Property | Value |
|---|---|---|
| 1. | General equation | x2⁄a2 - y2⁄b2 = 1 |
| 2. | Parametric equation | R(t) = a cosh(θ)î + b sinh(θ)ĵ |
| 3. | Co-ordinates of Center | (h,k) |
| 4. | Eccentricity | (e = c⁄a) > 1 |
| 5. | Linear Eccentricity | c = (a2 + b2)1⁄2 |
| 6. | Co-ordinates of Focii | (-c+h,k) and (c+h,k) |
| 7. | Equation of Major axis | y = k |
| 8. | Equation of Minor axis | x = h |
| 9. | Equation of Tangent at point (x0, y0) | (x0x)⁄a2 - (y0y)⁄b2 = 1 |
| 10. | Equation of Normal at point (x0, y0) | (a2x)⁄x0 + (b2y)⁄y0 = a2 + b2 |
| 11. | Equation of Directrix | x = a⁄e and x = -a⁄e |
| 12. | Length of Latus rectum | 2b2⁄a |
| 13. | Focal parameter | b2⁄c |
Labelled hyperbola
Hyperbolas in real life
Thermal power plant, Bathinda
Comet following hyperbolic path
Hyperbolic gears
Surface of earth intersects the cone of shockwave of a jet
Markings on sharped pencil are actually hyperbolae
A household lamp emits light in hyperbolic shape
Sundial
LORAN navigation
LORAN, short for long range navigation, was a hyperbolic radio navigation system developed in the United States during World War II. It was first used for ship convoys crossing the Atlantic Ocean, and then by long-range patrol aircraft, but found its main use on the ships and aircraft operating in the Pacific theater during World War II.
LORAN, short for long range navigation, was a hyperbolic radio navigation system developed in the United States during World War II. It was first used for ship convoys crossing the Atlantic Ocean, and then by long-range patrol aircraft, but found its main use on the ships and aircraft operating in the Pacific theater during World War II.
This two minute youtube video is a shortened version of the 1947 "LORAN for Ocean Navigation" filmstrip produced by the Coast Guard as a sales pitch to commercial shipping lines to adopt LORAN