Conic Sections- Circle, Parabola, Ellipse, Hyperbola
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The document provides an overview of conic sections, which are the curves obtained by the intersection of a plane with a cone. It details the types of conic sections: circles, ellipses, parabolas, and hyperbolas, along with their definitions and standard equations. Additionally, it describes the conditions that determine the type of conic section based on the angle of intersection and geometric properties.
Conic Sections- Circle, Parabola, Ellipse, Hyperbola
- 2. α β THE INTERSECTION OF A PLANE WITH A CONE, THE SECTION SO OBTAINED IS CALLED A CONIC SECTION V m Lower nappe Upper nappe Axis Generator l This is a conic section.
- 3. TYPES OF CONIC SECTIONS
- 4. CIRCLE A CIRCLE IS THE SET OF ALL POINTS ON A PLANE THAT ARE EQUIDISTANT FROM A FIXED POINT ON A PLANE. O P(x,y)
- 5. (h,k) C P(x,y) O (0,0) x² + y² = r² (x – h) ² + (y – k) ² = r² α β When β = 90°, the section is a circle Standard Equation General Equation
- 8. TYPES OF CONIC SECTIONS
- 9. ELLIPSE AN ELLIPSE IS THE SET OF ALL THE POINTS ON A PLANE, WHOSE SUM OF DISTANCES FROM TWO FIXED TWO REMAINS CONSTANT. P P P F F ¹ ³² ²¹
- 10. α β O (0,c) (0,-c) (-b,0) (b,0) (0,-a) (0,a) x² y² a² b² — —+ = 1—+ x² y² b² a² — = 1 (-c ,0) (c, 0) When α < β < 90°, the section is an ellipse Vertical Ellipse Horizontal Ellipse (0,-b) (0,b) (a,0)(-a,0)
- 12. .
- 13. TYPES OF CONIC SECTIONS
- 14. A PARABOLA IS THE SET OF ALL POINTS IN A PLANE THAT ARE EQUIDISTANT FROM A FIXED POINT A B V PARABOLA (VERTEX) F ( focus) 1 2 3 4O P 1 P2
- 15. α β F(a,0)O x=-a y² = 4ax X' X Y' Y F(-a,0) O x=+a y² = -4ax X' X Y' Y F(0,-a) O y = a x² = 4ay X' X Y' Y F(0,a) O y = -a x² = -4ay X' X Y' Y When α = β, the section is an parabola Horizontal Parabola Horizontal Parabola Vertical Parabola Vertical Parabola
- 18. TYPES OF CONIC SECTIONS
- 19. HYPERBOLA F ( focus)V (verte x) A B A HYPERBOLA IS THE SET OF ALL POINTS,THE DIFFERENCE OF WHOSE DISTANCES FROM TWO FIXED POINTS IS CONSTANT V (verte x) F ( focus)
- 20. 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 x² y² a² b² — —- = 1 - y² x² a² b² — —- = 1 When 0 ≤ β < α; the plane cuts through both the nappes & the curves of intersection is a hyperbola
- 23. Conic Section Standard Eq. General Eq. Circle x² + y² = r² (x – h) ² + (y – k) ² = r² Parabola y² = 4ax (y-k)² = 4a(x+h) Ellipse Hyperbola x² y² a² b² — —+ = 1 (x-h)² (y-k)² a² b² — + — = 1 x² y² a² b² — —- = 1 (x-h)² (y-k)² a² b² — - — = 1