source: ncert
topic: geometry
class_level: class_11
chapter: ch11
difficulty: intermediate
Class 11 | Ch11: Conic Sections CIRCLE: (x-h)^2+(y-k)^2=r^2. Centre (h,k), radius r. General: x^2+y^2+2gx+2fy+c=0. Centre (-g,-f), radius=sqrt(g^2+f^2-c). PARABOLA (standard forms): y^2=4ax: opens right. Focus (a,0). Directrix x=-a. Vertex (0,0). Axis=x-axis. y^2=-4ax: opens left. Focus (-a,0). Directrix x=a. x^2=4ay: opens up. Focus (0,a). Directrix y=-a. Axis=y-axis. x^2=-4ay: opens down. Focus (0,-a). Directrix y=a. Latus rectum: chord through focus perpendicular to axis. Length=4a. ELLIPSE: x^2/a^2+y^2/b^2=1 (a>b>0). Centre (0,0). Vertices (±a,0). Co-vertices (0,±b). Foci (±c,0) where c^2=a^2-b^2. Eccentricity e=c/a<1. Length of major axis=2a. Minor axis=2b. Latus rectum=2b^2/a. HYPERBOLA: x^2/a^2-y^2/b^2=1. Foci (±c,0) where c^2=a^2+b^2. Eccentricity e=c/a>1. Asymptotes: y=±(b/a)x. Latus rectum=2b^2/a. SOLVED EXAMPLES: Example 1: Parabola y^2=12x. Find focus, directrix, latus rectum. 4a=12 → a=3. Focus (3,0). Directrix x=-3. Latus rectum=12. Example 2: Ellipse, foci (±3,0), semi-major axis a=5. c=3, a=5. b^2=a^2-c^2=25-9=16. b=4. Equation: x^2/25+y^2/16=1. Example 3: Find equation of circle with centre (2,-3) and radius 5. (x-2)^2+(y+3)^2=25. Expanded: x^2+y^2-4x+6y-12=0. COMMON MISTAKES: Ellipse: c^2=a^2-b^2 (c<a). Hyperbola: c^2=a^2+b^2 (c>a). For parabola y^2=4ax: COMPARE with standard form to find a (4a=coefficient). Eccentricity: circle e=0, ellipse 0<e<1, parabola e=1, hyperbola e>1.