Conic Sections

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A conic section is a curve obtained by the intersection of a plane with a double-napped right circular cone. Depending on the angle and position of the intersecting plane, we get different types of cu

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A circle is formed when the intersecting plane is perpendicular to the axis of the cone (β = 90°). The plane cuts the nappe completely across one nappe of the cone, creating a circular cross-section.

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A circle is the set of all points in a plane that are equidistant from a fixed point called the center. The standard equation of a circle with center (h, k) and radius r is: (x - h)² + (y - k)² = r²

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When the center is at the origin (0, 0), the equation becomes: x² + y² = r²

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A parabola is the set of all points in a plane that are equidistant from a fixed point (called the focus) and a fixed line (called the directrix). If F is the focus and l is the directrix, then for an

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1. y² = 4ax (opens rightward, focus at (a,0), directrix x = -a) 2. y² = -4ax (opens leftward, focus at (-a,0), directrix x = a) 3. x² = 4ay (opens upward, focus at (0,a), directrix y = -a) 4. x² = -4a

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The latus rectum of a parabola is a line segment perpendicular to the axis of the parabola, passing through the focus, with endpoints on the parabola. For the parabola y² = 4ax, the length of the latu

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• If y² term: axis along x-axis. Opens right if coefficient of x is positive, left if negative. • If x² term: axis along y-axis. Opens up if coefficient of y is positive, down if negative. Example: y²