Revise: Circles and Parabolas
Conic sections, the general equations of circles and parabolas, and translation and rotation of axes.
Conics come from slicing a cone: circle, ellipse, parabola, hyperbola.
x²+y²=9 → circle. y=x²+3 → parabola.
Complete the square to recover standard form.
x²+y²−6x+4y−12=0 → center(3,−2), r=5.
Every point equidistant from focus and directrix.
Focus(0,2), directrix y=−2 → (4,2) checks out.
(x−h)²=4p(y−k), vertex (h,k).
Vertex(2,3), p=1 → (x−2)²=4(y−3).
A circle has matching x², y² coefficients; a parabola has only one.
Flashlight beam on a wall at different angles.
Add the same constants to both sides when completing the square.
x²+y²+2x−4y−4=0 → radius 3.
This defining property is what the algebraic equation is built from.
Reflective property used in satellite dishes.
Divide the y-coefficient by 4 to find p, not use it directly.
x²=12y → p=3.
The focus and directrix shift along with the vertex too.
(x−1)²=8(y+2) → vertex (1,−2).
Subtract (not add) when going from old to new coordinates.
New origin (2,5) → x'=7 for point (9,5).