Conics

Conic Sections

The distance between the focus and vertex is \(p\).

Every point on the parabola is equidistant from the focus to the directrix.

The latus rectum is a constant line that passes through the focus and parallel to the directrix. This line is \(4p\) units long.

\[ (x - h)^2 = 4p(y - k) \]

\[ y - k = \frac{(x - h)^2}{4p} \]

Remember \(y - k = a(x - h)^2\)

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Therefore, \(a = \frac{1}{4}p\)

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\[ (x - h)^2 + (y - k)^2 = r^2 \]

Co-vertices are part of the minor axis and vertices are part of the major axis.

Variables:

\(a\) - the center to a vertex
- \(a\) will always be the longest side
- Known as semi-major axis
\(b\) - the center to a co-vertex
- Known as semi-minor axis
\(c\) - the center to a focus
\(e\) - eccentricity describing the dilation of an ellipse

These variables will always form a right triangle, and thus can be written in Pythagorean Theorem.

\[ a^2 = b^2 + c^2 \]

\[ \frac{(x - h)^2}{b^2} + \frac{(y - k)^2}{a^2} = 1 \]

\[ \frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1 \]

Convert the equation into standard form. Then identify center, foci, eccentricity, and endpoints of major and minor axes.

\[ 16x^2 - 96x + 9y^2 + 36y + 36 = 0 \]

Complete the square

\[ 16(x - 3)^2 + 9(y + 2)^2 = 144 \]

Divide both sides by 16

\[ \frac{(x - 3)^2}{9} + \frac{(y + 2)^2}{16} = 1 \]

\(a\) is 16 since \(a\) must be the largest length. Therefore, this is a vertical ellipse.

Center: \((3, -2)\)

Vertices: \((3, -2 \pm 4)\)

Foci: \((3, -2 \pm \sqrt{7})\)

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