242 ANALYli aXOMXTBT [Cn. \. 



9. Tangenta to the ellipae 4r + 3y = 5 are inclined at 60 to the 

 tin.l the points of contact 



10. Fiixl the equation of an ellipse (center at the origin) of eccen- 

 tricity I 8uch that the subtangent for the point (:), V) w (- ) 



11. Fiii<l the chord of contact for tangents from the point ( '.. -J) to 

 the ellipse x* + 4y = 4. Find also the equation of the line from < 



to the middle point of this chord. 



12. Find tin- tangents to the ellipse 7x* + 8y* = 56 which make the 

 angle tan" 1 3 with the line x + y + 1 = 0. 



13. Find the product of the two segments into which a focal chord is 

 divided by the focus of an ellipse, using Art. 131. 



14. Find the equation of a tangent, and also of a normal, to the ellipse 

 :r* + 4 y s = 16, each parallel to the line 3 x 4 y = 5. 



15. Find the pole of the line 3 x - 4 y = 5 with reference to the ellipse 

 x* + 4 y* = 16; also the intercepts on the axes made by a line through UK; 

 ].].- and perpendicular to the polar. 



16. Find the points on the ellipse We 2 + ay = a*/> 2 , such that the tan- 

 gent makes equal (numerical) angles with the axes; such that the 

 subtangent equals the subnormal. 



146. Auxiliary circles. Eccentric angle. The circum- 

 scribed and inscribed circles for the ellipse (Fig. 107; 

 called auxiliary circles, and bear an important part in the 

 theory of the ellipse. Let the equation of tin* ellipse be 



The circle described on its major axis as diameter is called 

 th major auxiliary circle ; its equation is 



* + y = a; ... (2) 



and the circle on the minor axis as diameter in the minor 

 auxiliary circle ; its equation is 



6. ... (3) 



