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Ellipse

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THE ELLIPSE

An ellipse and a hyperbola are the locus of a point, which moves such that its distance from a fixed point (called the focus) is always e times its distance from a fixed line ( called the directrix). For an ellipse e<1 and for a hyperbola e1.

THE ELLIPSE

The equation of an ellipse in the standard form is given by

 

x2/a2 + y2/b2 = 1

The relation between a and b is given by

 

b2 = a2(1-e2)

  

  PROPERTIES OF AN ELLIPSE

  1. The points A and A’ are called the vertices of the ellipse. The coordinates of A and A’ are given by (-a,0) and (a,0) respectively. The line AA’ is called the major axis of the ellipse and its length is given by 2a.

  2. The line BB’ is called the minor axis of the ellipse. The coordinates of B and B’ are given by ( 0,b) and (0,-b). The length of the minor axis is given by 2b. The points B and B’ are the points where the y-axis intercepts the ellipse.

  3. The chord of the ellipse, which is perpendicular to the major axis and which also passes through the focus of the ellipse is called the latus rectum of the ellipse. It is represented by the equation

     

    X = -ae

    Its length is given by 2b2/a, and the coordinates of the end point are given by

    L = (-ae , b2/a) and L’ = (-ae , -b2/a)

  4. Since the ellipse is a symmetrical figure (as shown above ) there are two foci and two directrix. The coordinates of the second focus is given by (ae,0), and the equation of the second directrix is given by x = a/e.

  5. The focal distance of any point is defined as the distance between the focus and the point.

NOTE; the sum of the focal distances of any point is always equal to the length of the major axis.

From the figure SP + SP’ = 2a

6. The parametric coordinate of an ellipse is given by ( a cosq , b sinq ). This equation satisfies the equation of an ellipse for all values of q .

NOTE: The equation of an ellipse, for which the major axis lies along the y-axis and the minor axis on the x-axis is given by

 

X2/b2 + y2/a2 = 1

TANGENTS AND NORMALS

TANGENTS

The tangent at any point, A = (x1,y1), on an ellipse is given by the equation

 

x x1 /a2 + y y1/b2 = 1

The equation of the tangent in terms of slope is given by

 

y = mx +- (am2 +b2)1/2

NORMALS

The normal at any point, A = (x1,y1), on an ellipse is given by the equation

 

(x - x1 ) / x1/a2 + (y - y1)/ y1/b2 = 1

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