lecture #7 analytic geometry

Lecture #7EllipseParts of Ellipse and its graph• Equation of Ellipse

- Standard Equation- General Equation

• Formulas

ELLIPSEAn ellipse is defined by two points, each called a

focus. If you take any point on the ellipse, the sum of the distances to the focus points is constant.

PARTS OF AN ELLIPSEVertices – the points at which an ellipse makes its

sharpest turns and lies on the major axis, also end of major axis

Co-vertices – ends of minor axisFocus/foci – point/s that define the ellipse and lies on

the major axisMajor axis – the longest diameter of the ellipseMinor axis – the shortest diameter of the ellipse

EQUATIONS OF ELLIPSEThe equation of ellipse, if the center is at the origin,

major axis at x-axis, is given by

EQUATIONS OF ELLIPSEThe equation of ellipse, if the center is at the origin,

major axis at y-axis, is given by

EQUATIONS OF ELLIPSEThe equation of ellipse, if the center is at (h, k), major

axis parallel to x-axis, is given by

EQUATIONS OF ELLIPSEThe equation of ellipse, if the center is at (h, k), major

axis parallel to y-axis, is given by

The general equation of ellipse is given by

FORMULAS(if center is at the origin and major axis at x-

axis)

Vertices Co-vertices

(a, 0) (-a, 0) (0, b) (0, -b)

Foci Length of LR

(c, 0) (-c, 0)Length of major and minor axis

2a (major) 2b (minor)

Ends of Latera recta

FORMULAS(if center is at the origin and major axis at y-

axis)

Vertices Co-vertices

(0, a) (0, -a) (b, 0) (-b, 0)

Foci Length of LR

(0, c) (0, -c)Length of major and minor axis

2a (major) 2b (minor)

Ends of Latera recta

FORMULAS(if center is at (h, k) and major axis at x-axis)

Vertices Co-vertices

(h + a, k) (h - a, k) (h, k + b) (h, k-b)

Foci Length of LR

(h + c, k) (h - c, k)

Length of major and minor axis

2a (major) 2b (minor)

Ends of Latera recta

FORMULAS(if center is at (h, k) and major axis at y-axis)

Vertices Co-vertices

(h, k + a) (h, k - a) (h + b, k) (h - b, k)

Foci Length of LR

(h, k + c) (h, k - c)

Length of major and minor axis

2a (major) 2b (minor)

Ends of Latera recta

Sample ProblemGraph the ff. ellipses.