Ellipse Assignment Help

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Ellipse:

An ellipse is locus of a point which moves in the plane such that the ratio of the distances from a fixed point (called as focus) and from the fixed straight line (called as directrix) is always constant and less than 1. And this constant ratio is called as eccentricity of ellipse.

Standard equation of the ellipse

    1937_Ellipse1.png

here b2 = a2(1 - e2).

            The eccentricity of ellipse 922_Ellipse2.png can be given by the relation b2 = a2(1 - e2), i.e., e2 = 1 - b2/a2

 

 734_Ellipse.png

            An ellipse has 2 foci and 2 directrices.

Latus Rectum: Latus rectum is line which passes through the focus of ellipse and perpendicular to major axis. End points of latus rectum are given by(ae, b2/a) and (ae,-b2/a).The length of semi latus rectum can be given by b2 / a.

Focal Distance of a Point: To find distance of any point on the ellipse from the focus, we use definition of ellipse.

Let P(x, y) be a point on ellipse. Then

S¢P = ePN' = e(a/e -x) =  a - ex

SP = ePN = e ( a/e + x) = a + ex

S'P + SP = 2a

Þ The sum of the focal distances of any point on ellipse is equal to its major axis. Also SS'<SP+S'P=2a.

2117_Ellipse3.png

 

Other Forms of the Ellipse:

(i) If in equation 922_Ellipse2.png, a2< b2 ,  then  major and  minor axis  of ellipse  lie along  the  y and  x -axis and  are of  lengths 2b  and  2a respectively. The foci become (0, ± be) , and directrices become  y =  ± b/e 

      here  305_Ellipse5.png. The length of semi-latus rectum becomes a2/b.   

(ii)  If centre  of the  ellipse  be taken (h, k) and  axes  parallel  to x and y-axes,  then the  equation of ellipse is 1238_Ellipse6.png.

Example: Find out the equation of ellipse referred to its centre whose foci are points (4, 0) and (-4, 0) and whose eccentricity is 1/3.

Solution:       Let equation to the ellipse be  922_Ellipse2.png           ......(1)

            Distance between foci = 2ae = 4 + 4 = 8              ......(2)

            Putting value of e = 1/3 in (2), a = 4/e = 12

            Again b2 = a2(1 -e2) = 144 (1-1/9) = 128

            Put in (1) we have 2441_Ellipse7.png 

            => 8x2 + 9y2 = 1152

 

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