158 AKALYIK <,t:<>MKTRY . \ H. 



It' tills lilir |..i>srs through /'._,, thru 



j+ytfi + ffc^ + v + J'Cyi + yO+ff-o- . . 



But tlio equation of the polar of P f (Art. 9- 



and equation ( -\ ) proves that tin- locus of equation ( {) passes 



through J\, which cstahlislu-s tin- thruivm. 



EXERCISES 

 1. Find the polar of the point (6, 8) with reference to the circle 



2. Find the polar of the point (1, 2) with regard to the circle 



3. Find the pole of the line 4z + 6y = 7, and of the line ax + by-1 =0, 

 with regard to the circle or 2 -f y 8 = 3.1. 



4. Find the equations of the two tangents to the circle z 2 -f y 2 = 65 

 from the j>oint (4, 7); from the point (11, 3). 



5. Show that if the polar of (A, K) with respect to the circle z a -f y 2 = < 

 touch the circle 4 (z 2 + y 2 )= c 2 , then the pole (A, A) will lie on the circle 



6. Show that the pole of the line joining (5, 7) and (-11, 1 ) i 

 point of intersection of the polars of those two points with reference to 

 the circle x 2 -f y 2 = 100. 



7. Find the pole of the line 2z 3y = with respect to the circle 



8. Show what specialization of a polar converts it into a choi 

 contact, and what further specialization converts it into a tangent. 



94. Geometrical construction for the polar of a given point, 

 and for the pole of a given line, with regard to a given circle. 

 Since the relation between a polar and its pole (see <h t. 

 Art. 91) is independent of the coordinate axes, then : 

 the given circle may, without loss of generality, be assumed 

 to have its center at the origin. 



If PjS^j, yj) is any given point, and 



r 2 . . (1) 



