Jefferson's Scribe - Introduction to conics =3
YES I WAS CHOSEN TO BE FRIDAY'S SCRIBE BY NONE OTHER THAT MANNY!!! thanks pal. I don't know why but I'm really pumped on doing the scribe post. Alright its time to get to business shall we? Today was an interesting yet fun class today. we've met a new character in Mr K yesterday. Rurouni Sensei Kuropatwa was his name. (rorouni = wandering samurai. sensei = master/teacher in Japanese). He un sheathed his meter stick Katana blade(katana = Japanese sword) and asked the class what geometric shape it was. "Its a line Mr. K", says the class. And it was. Mr k grabbed his meter stick in the middle and started spinning it around really fast and then asked the class what shaped he was making. "Two cones Mr. K", says the class. And it was. He then took out his favourite cone (you know? the one that sits in the corner?). He then took his blade and sliced the top part off. It was so fast that the class couldn't see it. He then took off the top and told us that the cross view would be a circle. .He the cut the cone diagonally.(the slice was so fast that you couldn’t see it cutting through the cone). And out came a what looked like an oval or an ellipse. Then he cut it again at a steeper angle and out came a parabola. He then drew a diagram of a double napped cone and the cuts he made to it. Then he said that if you were to cut the cone perpendicular to it(vertically) it would make out a hyperbola. So a recap folks:
Circle- Starts a cone and made from a horizontal cut. At cross view it would be seen as a circle
Ellipse - found from cutting a cone in an angle. An ellipse is also a circle that someone sat on.
Parabola- cutting the base of the cone at a steep angle
Hyperbola - perpendicular to base
These are all called degenerates
Sensei K also said that a conic section has 4 ways of cutting. Those were the 4 ways I was mentioning about. He also said that if you also cut parallel to the vertex of the double napped cone it would be called a degenerate circle.(i'm not sure if that's what it's called)
We then had a new dictionary entry
There are 4 tipes of conic sections
The conic section arise by slicing a double napped cone and looking at the cross section
The “Degenerates”: A point horizontal slice through the vertices produces a point, a degenerate circle. A line: A slice along one of the edges produces a line, a degenerate ellipse. A pair of intersecting lines a perpendicular slice through the vertices produces a pair of intersecting lines ; a degenerate hyperbola.
Locus: A fixed point used in generating the locus for a conic section
Directrix: A fixed line used in generating the locuz for a conic section
Locus definition for conic sections:
A locus (set of points) such that the distance from a fixed point(the focus) divided by the distance from a fixed line(the directrix) is constant.
Definition: The locus of points determined by a fixed distance(the radius) and a fixed point(the centre)
Deriving the standard form for the equation of a circle
(line)OP = T [by definition]
Sqr[(x - h)^2+(y - k)^2] = T [distance formula]
(x + h)^2+(y - k)^2 = T^2[square both sides]
The standard form for the equation of a circle.
1. Write the equation of a circle with centre (-2, 3) and radius sqr(5)
(x - h)^2 + (y - k)^2 = y^2 h = -2 k = 3 r = sqr(5) (x - (-2))^2 + (y-(3))^2 = (sqr(5))^2 (x+2)^2 + (y- 3 )^2 = 5
2. Find the centre and radius of the circle (x - 3)^2 + (y - 5)^2 = 4
Centre (3, -5) Radius is sqr(4) = 2
3. Find the centre and radius of the circle x^2 + y^2 - 6x + 10y = 0
x^2 - 6x + y^2 + 10y = -9 x^2 - 6x +9 + y^2 +10 +25 = -9 +9 +25 (x-3)^2 + (y+5)^2 = 25
Centre(3, -5) Radius sqr(25)= 5
We also have homework:textbook, page 141 every 3 up to #63
And thus ends my scribe. I'M VERY PROUD OF MY WORK :")!!! well its time to pick monday's scribe 2 weeks from now. AND HIS NAME WILL BE CALVIN !!! Lets all give him a round of applause *applause*
And its time for me to leave. i've got a lot of things to do on my spring break list.
HAVE A NICE SPRING BREAK GUYS!!
YOU TOO MR. K!! =3