Analytic Geometry
Analytic geometry in NEB Class 12 covers straight lines and conic sections: circle, parabola, ellipse and hyperbola. You need to recognize standard forms, convert general equations by completing the square, and compute foci, eccentricity, directrices and tangents.
Core rules
- A line with slope m through (x₁,y₁): y - y₁ = m(x - x₁); general form ax + by + c = 0 has slope -a/b. Distance from (x₁,y₁): |ax₁ + by₁ + c|/√(a² + b²).
- Parallel lines have equal slopes; perpendicular slopes multiply to -1.
- Circle x² + y² + 2gx + 2fy + c = 0: centre (-g,-f), radius √(g² + f² - c).
- Parabola y² = 4ax: focus (a,0), directrix x = -a.
- Ellipse x²/a² + y²/b² = 1 with a > b: c² = a² - b², e = c/a, foci (±c,0).
- Hyperbola x²/a² - y²/b² = 1: c² = a² + b², e = c/a, asymptotes y = ±(b/a)x.
Worked example
For 9x² + 25y² = 225, divide by 225: x²/25 + y²/9 = 1. Thus a = 5, b = 3, c = √(25 - 9) = 4, e = 4/5. The foci are (±4,0), and the major axis lies along the x-axis.
Traps
- Reading a as a² in the denominators.
- Using a² - b² for a hyperbola; there it is a² + b².
- Forgetting to divide the general equation by the constant before identifying a conic.
- Omitting the absolute value in the distance formula.
- Assuming the major axis is always horizontal; compare denominators.