1. the ecentricity of an ellipse is 1/2, focus is S(0,0) and the major axis and directrix intersect at Z(-1,-1). find the coordinates of the center of the ellipse.

2. show that the sum of the focal distances of any point on the ellipse 9x² +16y² =144 is constant.

3.A point moves on a plane in such a manner that the sum of its distances from the points (5,0) and (-5,0) is always constant and equal to 26. show that the locus of the moving point is the ellipse x²/169+y² /144=1.

4. show that for an ellipse the straight line joining the upper end of one of the latus rectum and the lower end of other latus rectum passes through the center of the ellipse.

Q2.

Equation of ellipse by dividing both sides by 144,we get

Let the points in a plane  F1 = {(F,0)}  and  F2 = {(-F,0)}  be 
the foci of the ellipse where

Let  M = (x,y) be an arbitrary point in the ellipse. Consider the vectors   = F1M = (x-F,y)  and   = F2M = (x+F,y)

These vectors are called the  focal vectors  of the point  M  in the ellipse

=

Put

=

Similarly, by solving we will get    = 

therefore sum of focal distance =

+=8 constant

Q3.

Let E be an ellipse and Its equation is given by 

   (1) 

Given the points in a plane  F1 = {(5,0)}  and  F2 = {(-5,0)}  be 
the foci of the ellipse where 

 = . 

 (2)

Let  M = (x,y) be an arbitrary point in the ellipse. Consider the vectors   = F1M = (x-F,y)  and   = F2M = (x+F,y)
These vectors are called the  focal vectors  of the point  M  in the ellipse. 

the point M in the ellipse. We have 
=== =  (next substitute = since the point (x,y) satisfies (1)) 
=  =  =  =  = . 
Similarly,   =  =   . . . =   = . 
Thus,   +  =  +  = .

Given,  2a=26 Therefore a=13

From equation 2, we get 

Therefore, Equation of ellipse,