Analytic Geometry

Analytic Geometry

Parabolas

Analytic geometry of a parabola:

| | Equation | Vertex | Axis | Focus | Directrix | | ----------- | ---------------- | ------ | ---- | ------- | --------- | | Opens up | x^2=4py | (0,0) | x=0 | (0,p) | y=-p | | Opens down | x^2=-4py | (0,0) | x=0 | (0,-p) | y=p | | Opens right | y^2=4px | (0,0) | y=0 | (p,0) | x=-p | | Opens left | y^2=-4px | (0,0) | y=0 | (-p,0) | x=p | | Opens up | (x-h)^2=4p(y-k) | (h,k) | x=h | (h,k+p) | y=k-p | | Opens down | (x-h)^2=-4p(y-k) | (h,k) | x=h | (h,k-p) | y=k+p | | Opens right | (y-k)^2=4p(x-h) | (h,k) | y=k | (h+p,k) | x=h-p | | Opens left | (y-k)^2=-4p(x-h) | (h,k) | y=k | (h-p,k) | x=h+p |

If given an equation for a parabola like x^2 - 4x - 4y = 0:

  1. Check if you have an x^2 term or y^2 term
  2. Complete the square with respect to the squared term
  3. Arrange your equation to resemble one from the chart above
  4. Deduce the elements (pass in x=0 or y=0 to find intercept points)
x^2 - 4x - 4y = 0
x^2 - 4x = 4y    // -4/2 = -2 => (-2)^2 = 4
x^2 - 4x + 4 = 4y + 4
(x-2)^2 = 4y + 4
(x-2)^2 = 4(y+1)

If given information about a parabola and asked to find the equation:

  1. Sketch the graph
  2. Plug in what information you have
  3. Solve for the missing variables (like p)
  4. Simplify the original (solve for y)

Note: I need to find a better source for this other stuff

Polar equations

TODO

Ellipses

TODO -- similar to parabolas

Hyperbolas

TODO -- similar to parabolas

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