Conic Sections

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### Introduction to Conic Sections Conic sections are curves formed by the intersection of a plane and a double-napped cone. There are four main types: circle, ellipse, parabola, and hyperbola. Their properties are fundamental in physics, engineering, and astronomy. ### General Equation The general second-degree equation for a conic section is: $$Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$$ The type of conic section can be determined by the discriminant $B^2 - 4AC$: - If $B^2 - 4AC 0$: Hyperbola ### Circle A circle is the set of all points equidistant from a central point. - **Standard Equation:** $(x-h)^2 + (y-k)^2 = r^2$ - $(h, k)$: Center - $r$: Radius - **Properties:** - Eccentricity $e = 0$ - Directrix is at infinity - **Graph:** ### Ellipse An ellipse is the set of all points where the sum of the distances from two fixed points (foci) is constant. - **Standard Equation:** - Horizontal major axis: $\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1$ - Vertical major axis: $\frac{(x-h)^2}{b^2} + \frac{(y-k)^2}{a^2} = 1$ - $(h, k)$: Center - $a$: Semi-major axis length - $b$: Semi-minor axis length - $a^2 = b^2 + c^2$, where $c$ is the distance from the center to a focus. - **Properties:** - Foci: $(\pm c, 0)$ or $(0, \pm c)$ relative to the center - Vertices: $(\pm a, 0)$ or $(0, \pm a)$ relative to the center - Eccentricity $e = \frac{c}{a}$, $0 ### Parabola A parabola is the set of all points equidistant from a fixed point (focus) and a fixed line (directrix). - **Standard Equation:** - Opens horizontally: $(y-k)^2 = 4p(x-h)$ - Opens vertically: $(x-h)^2 = 4p(y-k)$ - $(h, k)$: Vertex - $p$: Distance from vertex to focus and vertex to directrix - **Properties:** - Focus: $(h+p, k)$ or $(h, k+p)$ - Directrix: $x = h-p$ or $y = k-p$ - Axis of symmetry: $y=k$ or $x=h$ - Eccentricity $e = 1$ - **Graph:** ### Hyperbola A hyperbola is the set of all points where the absolute difference of the distances from two fixed points (foci) is constant. - **Standard Equation:** - Horizontal transverse axis: $\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1$ - Vertical transverse axis: $\frac{(y-k)^2}{a^2} - \frac{(x-h)^2}{b^2} = 1$ - $(h, k)$: Center - $a$: Distance from center to vertex - $b$: Defines asymptotes - $c^2 = a^2 + b^2$, where $c$ is the distance from the center to a focus. - **Properties:** - Foci: $(\pm c, 0)$ or $(0, \pm c)$ relative to the center - Vertices: $(\pm a, 0)$ or $(0, \pm a)$ relative to the center - Asymptotes: $y-k = \pm \frac{b}{a}(x-h)$ (horizontal) or $y-k = \pm \frac{a}{b}(x-h)$ (vertical) - Eccentricity $e = \frac{c}{a}$, $e > 1$ - Directrices: $x = \pm \frac{a}{e}$ or $y = \pm \frac{a}{e}$ - **Graph:** ### Eccentricity ($e$) Eccentricity is a measure of how much a conic section deviates from being circular. It is defined as the ratio of the distance from any point on the conic to a focus to the distance from that point to its corresponding directrix. - Circle: $e = 0$ - Ellipse: $0 1$