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# Conic Section

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### Conic Section

1. 1. PRESENTATION CONIC SECTIONS OF BY:- Sahil Soni
2. 2. CONIC SECTIONS THE INTERSECTION OF A PLANE WITH A CONE, THE SECTION SO OBTAINED IS CALLED A CONNIC SECTION. α V m Lower nappe Upper nappe Axis l Generator
3. 3. SECTIONS OF CONE Plane Cone β α α V m Lower nappe Upper nappe Axis l Generator
4. 4. CIRCLE A CIRCLE IS THE SET OF ALL POINTS IN A PLANE THAT ARE EQUIDISTANT FROM A FIXED POINT IN THE PLANE. O P (x,y) Radius
5. 5. When β = 90°, the section is a circle (h,k) C P (x,y) O (0,0) x² + y ² = r ² (x – h) ² + (y – k) ² α β
6. 6. AN ELLIPSE IS THE SET OF ALL THE POINTS IN A PLANE, THE SUM OF WHOSE DISTANCES FROM TWO FIXED POINTS IN THE PLANE ELLIPSE P P P F F ¹ ³ ² ² ¹
7. 7. When α < β < 90°, the section is an ellipse α β O (0,c) (0,-c) (-b,0) (b,0) (0,-a) (0,a) Y x² y² a ² b ² — — + = 1 x² y² b ² a² — — + = 1 (-c ,0) (c, 0)
8. 8. PARABOLA A Parabola is the set of all points in a plane that are equidistant from a fixed point in the plane. Focus O X' X Y' Y • P (x,y) Directrix
9. 9. When β = α, the section is a parabola α β F (a,0) O x = -a y ² = 4ax X' X Y' Y F (-a,0) O x = +a y ² = -4ax X' X Y ' Y F (0,-a) O y = a x ² = 4ay X' X Y ' Y F (0,a) O y = -a x ² = -4ay X' X Y ' Y
10. 10. A hyperbola is the set of all points in a plane, the difference of whose distances from two fixed points in the plane is a constant. Transverse axis F Conjugate axis F Centre Vertex Vertex HYPERBOLA ² ¹
11. 11. When 0 ≤ β < α; the plane cuts through both the nappes & the curves of intersection is a hyperbola α β Transverse axis F Conjugate axis F (c ,0) (a ,0) ( -c ,0) (-a ,0) O F F (0 ,c) (0 ,a) (0 ,-c) (0 ,-a) O ¹ ¹ ² ²