Mathematics
Conic Sections
Conic sections are the curves you get by slicing a cone with a plane: circle, ellipse, parabola, hyperbola. Each has a standard equation form that makes its center, vertices, foci, and axes immediately readable.
How to Approach Conic Sections
Identify which conic
Look at the squared terms. Both x² and y² with the same sign and coefficient = circle. Same sign, different coefficients = ellipse. Opposite signs = hyperbola. Only one squared = parabola.
Convert to standard form
Complete the square in x and y separately. Group the constants. The result lets you read off center (h, k), a, b, and c — the defining parameters.
Extract key features
Vertices: where the curve crosses its axes. Foci: c units from center where c² = a² ± b² depending on the conic. Eccentricity e = c/a tells you the shape (0 = circle, ≈1 = parabola, >1 = hyperbola).
Frequently Asked Questions
Why is a parabola y = x²?+
That's the standard form when the focus is at (0, 1/4) and the directrix is y = -1/4. Every point on the curve is equidistant from the focus and directrix.
What's eccentricity?+
A measure of how 'stretched' a conic is. Circle: e = 0. Ellipse: 0 < e < 1. Parabola: e = 1. Hyperbola: e > 1. Orbits of planets are ellipses; comets often have e > 1.
Where do conics appear in physics?+
Planetary orbits (ellipses), projectile trajectories (parabolas), comet paths (hyperbolas for unbound), satellite dish reflectors (paraboloids), radar antennas (hyperboloids).
Related Topics
More step-by-step guides in Mathematics and adjacent subjects.
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