Conic Sections

Take a regular ice cream cone. Now cut it with a flat plate at different angles. The edge you see is a conic. Cut straight across, you get a circle. Tilt a little, you get an ellipse like a stretched circle. Tilt more so it is parallel to the side of the cone, you get a parabola. Cut more steep so it cuts both top and bottom cone, you get a hyperbola with two separate curves.

Why should we care? Planets move in ellipses. Satellite dishes are parabolas. Headlights use parabola focus. Hyperbola is used in navigation and cooling towers. All four come from one cone, only the angle changes. Eccentricity e tells you which one: e=0 circle, e between 0 and 1 ellipse, e=1 parabola, e greater than 1 hyperbola.

Circle is the simplest conic. Fix a point C, fix a distance r. All points that are exactly r away from C make a circle. Like you tie a goat with a rope to a pole, the goat walks around and makes a circle.

If centre is (h,k) and radius r, then any point (x,y) on circle satisfies (x-h)² + (y-k)² = r². If centre is at origin (0,0), it becomes x² + y² = r². To check if a point is inside, outside or on circle, put it in LHS and compare with r². Less means inside, more means outside.

Example: centre (2,-1) radius 5 → (x-2)² + (y+1)² =25. If you are given equation, you can read centre and radius directly. This is called standard form.

Sometimes equation is given expanded. Like x² + y² + 2gx + 2fy + c =0. This is general form. It still is a circle if x² and y² coefficients are equal and there is no xy term.

Centre is (-g,-f). Radius is √(g²+f²-c). So you just look at numbers. Example x²+y²-4x+6y-12=0. Here 2g=-4 so g=-2, 2f=6 so f=3, c=-12. Centre (2,-3), radius √(4+9+12)=√25=5.

Condition to be real circle: g²+f²-c greater or equal 0. If =0, it is a point circle. If less than 0, imaginary, no real points. This form helps when you are given three points on circle - put them in general form and solve for g,f,c.

Parabola is set of points that are equally far from a fixed point (focus) and a fixed line (directrix). Like you stand so that distance to a lamp post equals distance to a wall line. The path you walk is a parabola.

Simplest one: focus (a,0), directrix x=-a, vertex at origin. Equation is y²=4ax. Here a is distance from vertex to focus. If a positive, it opens right. If a is negative, it opens left. Similarly x²=4ay opens up, x²=-4ay opens down.

So there are four types, all same shape, only direction changes. Vertex is middle between focus and directrix. Axis is line through focus perpendicular to directrix. This shape is used in satellite dish - all rays parallel to axis reflect to focus.

Take parabola y²=4ax. Vertex (0,0), focus (a,0), directrix x=-a. Latus rectum is a line through focus, perpendicular to axis. It cuts parabola at two points. Its length is 4a. It is like the width of parabola at focus.

For y²=4ax, ends of latus rectum are (a,2a) and (a,-2a). So if a=2, width is 8. Focal distance = a. If you are given equation y²=12x, compare with 4a=12 so a=3. Then focus (3,0), directrix x=-3, latus rectum 12.

Worked example: y²=-8x → 4a=8 but negative so a=2 opens left. Focus (-2,0), directrix x=2, latus rectum 8. Ends (-2,4) and (-2,-4). Quick way to draw is mark vertex, focus, go up and down 2a from focus.

Fix two points F1 and F2. Tie a string of length 2a between them, keep string tight with a pencil and move pencil around. The shape you get is an ellipse. So sum of distances to two foci is constant =2a.

Ellipse has centre at middle of foci, major axis 2a (longest), minor axis 2b (shortest). If a greater than b, major along x-axis, equation x²/a² + y²/b² =1. Foci are at (±c,0) where c²=a²-b². Eccentricity e=c/a which is less than 1. Circle is special ellipse with e=0 where both foci meet at centre.

Example a=5,b=3 → c²=25-9=16 so c=4. Foci (±4,0), vertices (±5,0), minor ends (0,±3). String length 10. Real life: planet orbit, whispering gallery.

Standard ellipse x²/a² + y²/b²=1. If a greater than b, major on x. If b greater than a, major on y and equation is x²/b²+y²/a²=1 but we usually rename so a is still major. Latus rectum is line through focus perpendicular to major axis. Length is 2b²/a.

So for a=5,b=3, latus rectum =2*9/5=18/5=3.6. There are two such lines, one at each focus. Area = πab. Perimeter has no simple formula.

Worked: x²/25+y²/9=1 → a=5,b=3,c=4,e=4/5=0.8, foci (±4,0), LR 18/5. If x²/9+y²/25=1 then major vertical a=5,b=3 foci (0,±4) vertices (0,±5).

Hyperbola is opposite of ellipse. Fix two foci. Set of points where absolute difference of distances to foci is constant =2a. So |PF1-PF2|=2a. It makes two separate open curves, left and right.

Std equation x²/a² - y²/b²=1 opens left right. Transverse axis 2a along x, conjugate axis 2b along y. Centre origin. Vertices (±a,0), foci (±c,0) where c²=a²+b². So c greater than a always. Eccentricity e=c/a greater than 1. Asymptotes are lines y=±(b/a)x that hyperbola approaches far away.

Example x²/9 - y²/16=1 → a=3,b=4,c²=9+16=25 c=5 e=5/3≈1.67. Latus rectum length 2b²/a=32/3≈10.67 ends (5,±16/3). Used in cooling towers and navigation.

Circle: Wheels, coins, plates, clock, manhole cover, steering. Any time you need equal distance from centre, circle comes. Park fountain spray is circular. Compass draws circle.

Parabola: Torch, bike headlight, car headlight, satellite dish, solar cooker. Shape takes parallel rays and sends to focus, or takes source at focus and sends parallel beam. That is why dish antenna at your home is parabola facing sky. Cricket ball path is parabola.

Ellipse: Planet orbits are ellipses with Sun at one focus. Whispering gallery in St Paul's - you whisper at one focus, friend hears at other focus. Stadium design, table with elliptical shape, egg shape is close to ellipse. Medical lithotripter breaks kidney stones using ellipse reflection property - waves from one focus go to stone at other focus.

Hyperbola: Cooling towers of power plants are hyperbola shape for strength and airflow. Navigation system LORAN uses difference of distances property of hyperbola. Sonic boom cone cuts ground as hyperbola. Telescope uses hyperbola mirrors to focus light.