1st; ANALYTIC GKOMKTRT [On. VIM. 



3. given th- forus at the point (0, 1), tli.- .'.ju.-ition of tln> directrix 

 y - 25 = 0, and the eccentricity ; 



4. given the center at the origin, and the semi-axes V2, V5. Fiinl 

 also the latus rectum. 



Fiti.l tho equation of an ellipse referred to its center, whose axes are 

 the coordinate axes, and 



5. \\ Inch passes through the two points (2, 2) and (3, 1). 



6. whose foci are the points (3, 0), (3,0), and eccentricity f 



7. whose foci are the points (0, 6), (0, ~0), and eccentricity f . 



8. whose latus rectum is 5, and eccentricity f. 



9. whose latus rectum is 8, and the major axis 10. 



10. whose major axis is 18, and which passes through the point 6, 4. 



Draw the following ellipses, locate their foci, and find their equations : 



11. given the center at the point ( .;. -j), the semi-axes 4 and 3, and 

 the major axis parallel to the x-axia (cf. Art. 112) ; 



12. given the center at the point (~8, 1), the semi-axes 2 and 5, ami 

 the major axis parallel to the y-axis ; 



13. given the center at the point (0, 7), the origin at a vertex, and 

 (2, 3) a point on the curve ; 



14. given the circumscribing rectangle, whose sides are the lines 

 a-4-l=0, 2 z - 3 = 0, y + 6 = 0, 3 y + 4 = ; the axes of the curve 

 being parallel to the coordinate axes. 



15. If b becomes more and more nearly equal to a, what curve does 

 the ellipse approach as a limit? 



113. Every equation of the form Ax* + By- + 2G& + 2 /'// 

 + C = O, in which A and B have the same sign, represents 

 an ellipse whose axes are parallel to the coordinate axes. 

 K I nations [44], [45], and [4(1], obtained for the ellipse, are 

 all, when expanded, of the form 



Ax*+By*+Z0x+ 5>J> + <7=0, . . . (1) 



where A and B have the same sign, and neither of them is zero. 

 Conversely, an equation of this form represents an ellipse 



