One day, a Russian student walked into the forest...
And walked into Baba Yaga!
Teach me about Conics, and I will let you pass
And she said...
Ok, the first thing you need to know is circles
Ok, the first thing you need to know is circles. Circles are a special case of an ellipse, where the ellipse turns into a circle. A basic formula is x²+y²=r²This is much like the distance formula, which comes from the Pythagorean theorem.
An easy example is x²+y²=9This means that because x and y are squared, 9 is the square of the radius. that means the radius is 3 units away from the center of the circle, in this case, 0,0 on a graph.
A harder equation is \s {x}-0^{2}+y-0^{2}=3. This may look harder but if you square and simplify both sides, you end up with x²+y²=9, much like the first equation. because of the equation, you know its a circle
The next Conic you must know is an ellipse. unlike a circle, an ellipse does not have an equal distance to the center, and it may look more like an oval
The most basic equation for an ellipse is \frac{x^{2}}{a^{2}}+\frac{x^{2}}{b^{2}}=1 a and b could be any numbers, while x and y will stay as variables
A more complex question would look like \frac{x+1^{2}}{49}+\frac{y-9^{2}}{121}=1Although this looks complicated, you can simplify. This lets you find the vertices and foci, and because it is not a perfect circle, you know it is an ellipse
Another important conic to learn is a parabola. a parabola does not meet itself like a circle or ellipse, but still curves. the standard form of a parabola is (y - k)2 = 4p(x - h)
I will walk you through a simple question, y=−2x2 .x2=4py.x2=−12y.The focus of the parabola which is in standard form x2=4py , is (0,p) .The directrix of the parabola which is in standard form x2=4py , is y=−p .So, the result of the equation is y=18 .
A parabola will always have a focus, a point in which the curve is equidistant on both sides. This is very importnad in finding the correct solution.