Algebra and Trigonometry

# Key Concepts

### 12.1The Ellipse

• An ellipse is the set of all points$( x,y ) ( x,y )$in a plane such that the sum of their distances from two fixed points is a constant. Each fixed point is called a focus (plural: foci).
• When given the coordinates of the foci and vertices of an ellipse, we can write the equation of the ellipse in standard form. See Example 1 and Example 2.
• When given an equation for an ellipse centered at the origin in standard form, we can identify its vertices, co-vertices, foci, and the lengths and positions of the major and minor axes in order to graph the ellipse. See Example 3 and Example 4.
• When given the equation for an ellipse centered at some point other than the origin, we can identify its key features and graph the ellipse. See Example 5 and Example 6.
• Real-world situations can be modeled using the standard equations of ellipses and then evaluated to find key features, such as lengths of axes and distance between foci. See Example 7.

### 12.2The Hyperbola

• A hyperbola is the set of all points$( x,y ) ( x,y )$in a plane such that the difference of the distances between$( x,y ) ( x,y )$and the foci is a positive constant.
• The standard form of a hyperbola can be used to locate its vertices and foci. See Example 1.
• When given the coordinates of the foci and vertices of a hyperbola, we can write the equation of the hyperbola in standard form. See Example 2 and Example 3.
• When given an equation for a hyperbola, we can identify its vertices, co-vertices, foci, asymptotes, and lengths and positions of the transverse and conjugate axes in order to graph the hyperbola. See Example 4 and Example 5.
• Real-world situations can be modeled using the standard equations of hyperbolas. For instance, given the dimensions of a natural draft cooling tower, we can find a hyperbolic equation that models its sides. See Example 6.

### 12.3The Parabola

• A parabola is the set of all points$( x,y ) ( x,y )$in a plane that are the same distance from a fixed line, called the directrix, and a fixed point (the focus) not on the directrix.
• The standard form of a parabola with vertex$( 0,0 ) ( 0,0 )$and the x-axis as its axis of symmetry can be used to graph the parabola. If$p>0, p>0,$ the parabola opens right. If$p<0, p<0,$ the parabola opens left. See Example 1.
• The standard form of a parabola with vertex$( 0,0 ) ( 0,0 )$and the y-axis as its axis of symmetry can be used to graph the parabola. If$p>0, p>0,$ the parabola opens up. If$p<0, p<0,$ the parabola opens down. See Example 2.
• When given the focus and directrix of a parabola, we can write its equation in standard form. See Example 3.
• The standard form of a parabola with vertex$( h,k ) ( h,k )$and axis of symmetry parallel to the x-axis can be used to graph the parabola. If$p>0, p>0,$ the parabola opens right. If$p<0, p<0,$ the parabola opens left. See Example 4.
• The standard form of a parabola with vertex$( h,k ) ( h,k )$and axis of symmetry parallel to the y-axis can be used to graph the parabola. If$p>0, p>0,$ the parabola opens up. If$p<0, p<0,$ the parabola opens down. See Example 5.
• Real-world situations can be modeled using the standard equations of parabolas. For instance, given the diameter and focus of a cross-section of a parabolic reflector, we can find an equation that models its sides. See Example 6.

### 12.4Rotation of Axes

• Four basic shapes can result from the intersection of a plane with a pair of right circular cones connected tail to tail. They include an ellipse, a circle, a hyperbola, and a parabola.
• A nondegenerate conic section has the general form$A x 2 +Bxy+C y 2 +Dx+Ey+F=0 A x 2 +Bxy+C y 2 +Dx+Ey+F=0$where$A,B A,B$and$C C$are not all zero. The values of$A,B, A,B,$ and$C C$determine the type of conic. See Example 1.
• Equations of conic sections with an$xy xy$term have been rotated about the origin. See Example 2.
• The general form can be transformed into an equation in the$x ′ x ′$and$y ′ y ′$coordinate system without the$x ′ y ′ x ′ y ′$term. See Example 3 and Example 4.
• An expression is described as invariant if it remains unchanged after rotating. Because the discriminant is invariant, observing it enables us to identify the conic section. See Example 5.

### 12.5Conic Sections in Polar Coordinates

• Any conic may be determined by a single focus, the corresponding eccentricity, and the directrix. We can also define a conic in terms of a fixed point, the focus$P(r,θ) P(r,θ)$at the pole, and a line, the directrix, which is perpendicular to the polar axis.
• A conic is the set of all points$e= PF PD , e= PF PD ,$ where eccentricity$e e$is a positive real number. Each conic may be written in terms of its polar equation. See Example 1.
• The polar equations of conics can be graphed. See Example 2, Example 3, and Example 4.
• Conics can be defined in terms of a focus, a directrix, and eccentricity. See Example 5 and Example 6.
• We can use the identitiesandto convert the equation for a conic from polar to rectangular form. See Example 7.