Introduction to Conic Sections
Conic sections are curves formed by intersecting a plane with a double cone, resulting in a circle, ellipse, parabola, or hyperbola. These shapes are fundamental in mathematics, physics, and engineering.
General Conic Equation:
\[ Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 \]
Discriminant: \( B^2 - 4AC \)
- \( B^2 - 4AC < 0 \): Ellipse (or circle if \( A = C \))
- \( B^2 - 4AC = 0 \): Parabola
- \( B^2 - 4AC > 0 \): Hyperbola
Circle
A circle is the set of points equidistant from a fixed point (center).
Standard Form:
\[ (x - h)^2 + (y - k)^2 = r^2 \]
Where: \( (h, k) \) is the center, \( r \) is the radius.
General Form:
\[ x^2 + y^2 + Dx + Ey + F = 0 \]
All Formulas:
- Area: \( A = \pi r^2 \)
- Circumference: \( C = 2\pi r \)
- Center (from general form): \( \left(-\frac{D}{2}, -\frac{E}{2}\right) \)
- Radius (from general form): \( r = \sqrt{\frac{D^2 + E^2 - 4F}{4}} \)
Ellipse
An ellipse is the set of points where the sum of distances to two foci is constant.
Standard Form (Centered at Origin):
\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \quad (a > b, \text{horizontal major axis}) \]
\[ \frac{x^2}{b^2} + \frac{y^2}{a^2} = 1 \quad (a > b, \text{vertical major axis}) \]
Standard Form (Centered at \( (h, k) \)):
\[ \frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1 \quad (\text{horizontal}) \]
\[ \frac{(x - h)^2}{b^2} + \frac{(y - k)^2}{a^2} = 1 \quad (\text{vertical}) \]
General Form:
\[ Ax^2 + Cy^2 + Dx + Ey + F = 0 \quad (A, C > 0, B^2 - 4AC < 0) \]
All Formulas:
- Eccentricity: \( e = \sqrt{1 - \frac{b^2}{a^2}} \quad (0 < e < 1) \)
- Foci (horizontal): \( (\pm c, 0) \), where \( c = \sqrt{a^2 - b^2} \)
- Foci (vertical): \( (0, \pm c) \)
- Area: \( A = \pi a b \)
- Perimeter (approximate): \( P \approx \pi \sqrt{2(a^2 + b^2)} \)
Parabola
A parabola is the set of points equidistant from a focus and a directrix.
Standard Form (Vertex at Origin):
\[ y^2 = 4ax \quad (\text{opens right}) \]
\[ y^2 = -4ax \quad (\text{opens left}) \]
\[ x^2 = 4ay \quad (\text{opens up}) \]
\[ x^2 = -4ay \quad (\text{opens down}) \]
Standard Form (Vertex at \( (h, k) \)):
\[ (y - k)^2 = 4a(x - h) \quad (\text{horizontal axis}) \]
\[ (x - h)^2 = 4a(y - k) \quad (\text{vertical axis}) \]
General Form:
\[ Ax^2 + Dx + Ey + F = 0 \quad \text{or} \quad Cy^2 + Dx + Ey + F = 0 \quad (B^2 - 4AC = 0) \]
All Formulas:
- Focus (for \( y^2 = 4ax \)): \( (a, 0) \)
- Focus (for \( x^2 = 4ay \)): \( (0, a) \)
- Directrix (for \( y^2 = 4ax \)): \( x = -a \)
- Directrix (for \( x^2 = 4ay \)): \( y = -a \)
- Latus Rectum: \( L = 4a \)
- Eccentricity: \( e = 1 \)
Hyperbola
A hyperbola is the set of points where the absolute difference of distances to two foci is constant.
Standard Form (Centered at Origin):
\[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \quad (\text{horizontal transverse axis}) \]
\[ \frac{y^2}{a^2} - \frac{x^2}{b^2} = 1 \quad (\text{vertical transverse axis}) \]
Standard Form (Centered at \( (h, k) \)):
\[ \frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1 \quad (\text{horizontal}) \]
\[ \frac{(y - k)^2}{a^2} - \frac{(x - h)^2}{b^2} = 1 \quad (\text{vertical}) \]
General Form:
\[ Ax^2 + Cy^2 + Dx + Ey + F = 0 \quad (B^2 - 4AC > 0) \]
All Formulas:
- Eccentricity: \( e = \sqrt{1 + \frac{b^2}{a^2}} \quad (e > 1) \)
- Foci (horizontal): \( (\pm c, 0) \), where \( c = \sqrt{a^2 + b^2} \)
- Foci (vertical): \( (0, \pm c) \)
- Asymptotes (horizontal): \( y = \pm \frac{b}{a}x \)
- Asymptotes (vertical): \( y = \pm \frac{a}{b}x \)
- Vertices (horizontal): \( (\pm a, 0) \)
- Vertices (vertical): \( (0, \pm a) \)