| --- |
| source: ncert |
| topic: geometry |
| class_level: class_11 |
| chapter: ch11 |
| difficulty: intermediate |
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| |
| 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. |
