Conic Sections

• The eccentricity of the ellipse (commonly denoted as either e or ) is
•
Directrix (2nd Definition Of Ellipse)
Directrix(2nd Definition Of Ellipse)
• Each focus F of the ellipse is associated with a line parallel to the minor axis called a directrix. Refer to the illustration on the right, in which the ellipse is centered at the origin. The distance from any point P on the ellipse to the focus F is a constant fraction of that point's perpendicular distance to the directrix, resulting in the equality e = PF/PD. The ratio of these two distances is the eccentricity of the ellipse. This property (which can be proved using the Dandelin spheres) can be taken as another definition of the ellipse. Besides the well-known ratio e = f/a, where f is the distance from the center to the focus and a is the distance from the center to a vertex (most sharply curved point of the ellipse), it is also true that e = a/d, where d is the distance from the center to the directns Of The Ellipse
• In this ellipse, constant value 2a must be bigger than 2c(focal length) • If focal points are on Y-axis, , b² >a² ,in other words, if focal points are on x or y axis, then x²or y²will divide a bigger positive number a²or b²
Focus Triangles
• Triangle PFE（F，E are focal points, P is on The Ellipse ) • Angle FPE is equal to A, then Area of Triangle PFE is b²multiplied by tan(A/2)
Focal Points
• Focus • The distance from the center C to either focus is f = ae, which can be expressed in terms of the major and minor radii:
Eccentricity
Equations
• Equations: • This means any noncircular ellipse is a squashed circle. If we draw an ellipse twice as long as it is wide, and draw the circle centered at the ellipse's center with diameter equal to the ellipse's longer axis, then on any line parallel to the shorter axis the length within the circle is twice the length within the ellipse. So the area enclosed by an ellipse is easy to calculate—it's the lengths of elliptic arcs that are hard. (in this equation, a must be bigger than b)
Different Types of Parabola
• • • • • • • • 1.y² =2px(p>0) Focal point (p/2,0) Directrix: x=-p/2 2.y² =-2px(p>0) Focal point (-p/2,0) Directrix: x=p/2 3.x² =2py(p>0) Focal point (0,p/2) Directrix: y=-p/2 4.x² =-2py(p>0) Focal point (0,-p/2) Directrix: y=p/2
Standard Formula
• 1.When focal point is on X-Axis • X² /a²- y² /b²= 1 （a>0，b>0,a may be larger than b, or smaller than b, even be equal） • 2.When focal point is on Y-Axis • y² /a²- x² /b²= 1 （a>0，b>0, also a may be smaller than b, or smaller than b, even be equal.） • 3.c² =a² +b² ,c is the half focal length.
Hyperbola and Its Conjugate Hyperbola
Focus Triangles
• Two focal points: F(-c,0),G(c,0),A is on hyperbola. Angle FAG is θ, then the area of the triangle FGA is b² cot θ.
Asymptote and Axis (use X² /a²y² /b²= 1 as an example)
• • • • • • • • X² /a²- y² /b²= 1 Order y=0: A (a,0), B(-a,0) AB Real Axis x=0:y² =-b² , no real root but we still draw this on the Y-axis C (0,b), D(0,-b) CD Imaginary Axis Asymptote: x/a ± y/b=0 Conjugate Hyperbola X² /a²- y² /b²= -1
Circular Directrix
• The ellipse can also be defined as the set of points that are equidistant from one focus and a circle, the directrix circle, that is centered on the other focus. The radius of the directrix circle equals the ellipse's major axis, so the focus and the entire ellipse are inside the directrix circle.
Focal Radius Of Ellipse
• If point P is on Ellipse ,focal points are A(-c,0) and B(c,0),P(x,y), • PA=a-ex, PB=a+ex • (Left Minus and Right Plus)
Parameter Formula For Ellipse
Parametric Formula of Hyperbola
• X=a sec θ • Y=b tan θ • (note : sec θ=1/cos θ)
Ellipse And Hyperbola
Chapter 3: Parabola
Illustration Of Directrix
• F is the focal point, L is the directrix
Definition of Ellipse
• Definition • In Euclidean geometry, the ellipse is usually defined as the bounded case of a conic section, or as the set of points such that the sum of the distances to two fixed points (the foci) is constant. The ellipse can also be defined as the set of points such that the distance from any point in that set to a given point in the plane (a focus) is a constant positive fraction less than 1 (the eccentricity) of the perpendicular distance of the point in the set to a given line (called the directrix). Yet another equivalent definition of the ellipse is that it is the set of points that are equidistant from one point in the plane (a focus) and a particular circle, the directrix circle (whose center is the other focus).