Analytic Geometry
The Analytic Geometry domain connects algebra and shape: lines, distance and midpoint, circles, parabolas, ellipses, and hyperbolas described by equations. Questions typically supply a general second-degree equation and ask for center, radius, vertex, foci, axes, or asymptotes - or reverse the process and build an equation from described features.
Core rules
- Slope m = (y2 - y1)/(x2 - x1) with point-slope form y - y1 = m(x - x1). Parallel lines share a slope; perpendicular slopes are negative reciprocals, so m1 · m2 = -1.
- Distance between points is sqrt((x2 - x1)^2 + (y2 - y1)^2); the midpoint averages the coordinates.
- Circle: (x - h)^2 + (y - k)^2 = r^2 has center (h, k) and radius r.
- Parabola: y = a(x - h)^2 + k has vertex (h, k); x = a(y - k)^2 + h opens horizontally, and the focus lies a distance 1/(4a) from the vertex.
- Ellipse: (x - h)^2/a^2 + (y - k)^2/b^2 = 1 with a > b has c^2 = a^2 - b^2, and the major axis follows the larger denominator.
- Hyperbola: x^2/a^2 - y^2/b^2 = 1 has c^2 = a^2 + b^2 and asymptotes y = ±(b/a)x through the center.
- Always complete the square and divide until the right side is 1 before naming any feature.
Worked example
Classify and describe x^2 + y^2 - 6x + 10y + 18 = 0. Group the variables: (x^2 - 6x) + (y^2 + 10y) = -18. Complete both squares, adding the same constants to the right: (x^2 - 6x + 9) + (y^2 + 10y + 25) = -18 + 9 + 25 = 16. This gives (x - 3)^2 + (y + 5)^2 = 16, a circle with center (3, -5) and radius 4.
Traps that cost marks
- Sign reversals: (x - 3)^2 means the center x-coordinate is +3, while (y + 5)^2 means it is -5.
- Mixing the focus relations, using c^2 = a^2 + b^2 for an ellipse or c^2 = a^2 - b^2 for a hyperbola.
- Forgetting to add the completed-square constants to both sides of the equation.
- Choosing the major axis by the value of a rather than by which denominator is larger and which variable it belongs to.
- Calling the perpendicular slope of 2 equal to 1/2 instead of -1/2.