Conic Sections
Circle, Parabola, Ellipse, Hyperbola
1. General Equation
$$ Ax^2 + 2Hxy + By^2 + 2Gx + 2Fy + C = 0 $$
| Conic | Condition ($$ \Delta \neq 0 $$) | Eccentricity (e) |
|---|---|---|
| Circle | $$ A = B, H = 0 $$ | $$ e = 0 $$ |
| Parabola | $$ H^2 = AB $$ | $$ e = 1 $$ |
| Ellipse | $$ H^2 < AB $$ | $$ 0 < e < 1 $$ |
| Hyperbola | $$ H^2 > AB $$ | $$ e > 1 $$ |
2. Circle
- Standard Form: $$ x^2 + y^2 = r^2 $$
(Center: (0,0), Radius: r) - Central Form: $$ (x-h)^2 + (y-k)^2 = r^2 $$
(Center: (h,k), Radius: r) - General Form: $$ x^2 + y^2 + 2gx + 2fy + c = 0 $$
Key Formulas:Center: $$ (-g, -f) $$
Radius: $$ \sqrt{g^2 + f^2 – c} $$
3. Parabola ($$ y^2 = 4ax $$)
| Parameter | Right Handed ($$ y^2=4ax $$) | Upward ($$ x^2=4ay $$) |
|---|---|---|
| Focus | $$ (a, 0) $$ | $$ (0, a) $$ |
| Directrix | $$ x = -a $$ | $$ y = -a $$ |
| Latus Rectum | $$ 4a $$ | $$ 4a $$ |
4. Ellipse ($$ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 $$)
Assuming $$ a > b $$ (Horizontal Ellipse)
Vertices: $$ (\pm a, 0) $$
Foci: $$ (\pm ae, 0) $$
Directrices: $$ x = \pm \frac{a}{e} $$
Important Relations:
$$ b^2 = a^2(1 – e^2) $$
Latus Rectum: $$ \frac{2b^2}{a} $$
5. Hyperbola ($$ \frac{x^2}{a^2} – \frac{y^2}{b^2} = 1 $$)
Vertices: $$ (\pm a, 0) $$
Foci: $$ (\pm ae, 0) $$
Directrices: $$ x = \pm \frac{a}{e} $$
Important Relations:
$$ b^2 = a^2(e^2 – 1) $$
Latus Rectum: $$ \frac{2b^2}{a} $$