ANM.Y il< 9XOM1 TMI 



EXERCISES 



1. i <li;i\\n to an ellipse from any external 

 point subtend equal angles at the focus. 



2. Kach of the two tangents drawn to the ellipse from a point 01 

 directrix subtends a right angle at the focus. 



3. A focal chord is perpendicular to the line joining its pole to the 

 focus. Show that this is also true for a parabola. 



4. The rectangle formed by the perpendiculars from the foci upon any 

 tangent is constant; it is equal to the square of the semi-minor-axis. 



5. The circle on any focal distance as diameter touches the major 

 auxiliary circle. 



6. The perpendicular from the focus upon any tangent, and the line 

 joining the center to the point of contact, meet upon the directrix. 



7. The perpendicular from either focus, upon the tangent at any point 

 ot the major auxiliary circle, equals the distance of the corresponding 

 point of the ellipse from that focus. 



8. The latus rectum is a third proportional to the major and minor 

 axes. 



9. The area of the ellipse is irab. 



SUGGESTION. Employ the fact, proved in Art. 146, that the ordinate 

 of an ellipse is to the corresponding ordinate of the major auxiliary 

 circle as b : a, and thus compare the area of the ellipse with that of its 

 major auxiliary circle. 



152. Diameters. As already shown in Articles 129 and 

 139, the definition of a diameter as the locus of the mid<llr 

 points of a system of parallel chords leads directly to its 

 equation. 



Let TO be the slope of the given system x>f parallel chords 

 of the ellipse whose equation is 



j.^-1 fV 



2 ' 12 - 1 * ' ' 



and let v = 7WX4- c 



