Analytic Geometry and Conic Sections
Analytic geometry and conic sections
Master the line forms and the standard equations of circles, ellipses, hyperbolas, and parabolas, including centers, vertices, foci, directrices, asymptotes, and eccentricity. For an ellipse x²/a²+y²/b²=1 with a>b>0, c²=a²−b² and e=c/a; for a hyperbola x²/a²−y²/b²=1, c²=a²+b² and the asymptotes are y=±(b/a)x. A parabola y²=4px has focus (p,0) and directrix x=−p. Intersections with a line usually lead to a quadratic, so Vieta's formulas and the discriminant are central.
Traps: putting the major axis on the wrong axis; using a²=b²+c² for a hyperbola; forgetting that a line through a focus can meet a conic in two points; and neglecting the case of a vertical line when using slope-intercept form. Always check whether the discriminant condition matches the geometric requirement (two points, tangency, or no intersection).
Worked example: for x²/25+y²/9=1, a=5, b=3, so c=4, foci are (±4,0), and e=4/5. For y²=8x, 4p=8 gives p=2, so the focus is (2,0) and the directrix is x=−2. Rule of thumb: identify the axis first, compute c from the correct relation, then translate the question into an equation or a Vieta condition.