258 GEOMETRY [Cn \ 



therefore 



H.-noe <h = 45 or 135 



for the extremities of equi-con jugate diameters, and the extremities are 



/*(*,.*- 4 *.-(-* *4- 



The equations of these diameters are 



y = z, and y = -- z. 

 a a 



Evidently these lines are the diagonals of the rectangle formed on the 

 axes of the curve. 



By Art. 155, (y), the length of each equi-con jugate semi-diameter is 



EXERCISES 



1. Find the diameter of the ellipse ~- 4- ^ = 1 which bisects the 

 chords parallel to the line 3 z -f 5 y + 7 = 0. 



2. Find the diameter conjugate to that of exercise 1. 



3. Show that the lines 2x - y = 0, x + 3y = are conjugate diame- 

 ters of the ellipse 2x* + 3y* = 4. 



4. For the ellipse 6 2 z 2 -f a 2 y 2 = a 1 ^ 2 , write the equations of diameters 

 conjugate to the line 



(a) ax = %, (ft) bx = ay. 



5. Prove that the angle between two conjugate diameters is a 

 maximum when they are equal. 



6. Show that the pair of diameters drawn parallel to the chords joii.- 

 ing the extremities of the axes are equal and conjugate. 



7. What are the equations of the pair of equi-con jugate diameters 

 of the ellipse 16y -f 9 z* = 1 1 1 V 



8. Two conjugate diameters of the ellipse + ^- = 1 have the 

 dopes } and f, respectively; find their lengths. 



9 (iivon the ellipse *+5y* = 5, find tin- eccentric angle for the 

 point whose abscissa is 1. Also find the diameter conjugate to the one 

 passing through this point 



