106 .i \ i/ TOO './>////.>- i vn. 



9. Find the f the circle passing through the origin ;ml 



the point (r,, /,), and having iU center on the y-axis. 



10. The point (3, -.'i) bisects a chord of tho circle x* -f y = 277 ; find 

 the equation of that chord. 



11. A < in In touches tho line 4ar -f <*y + S = at tho point (~ 

 nn<l paes through the point (">, 0); find its equation. 



12. A riivle, whose center coincides with tho origin, touches the lino 

 7 x - 1 1 y -f i> = ; find its equation. 



13. At the points in which the circle z* + y* - ax - ly = cuts tho 

 axes, tangents are drawn ; find the equations of these tangents. 



14. A circle, whose radius is V74, touch'> tin- line 5y = 1 x 1 at 

 the point (8, 11); find tin- equ tion of this circle. 



15. A circle is inscribed in the triangle (3, ~J), (y, 3), (3, 3): 



Its equation; find also the equations of the polars of tin- three vertices 

 with regard to this circle. 



16. Through a fixed point (ar,, yj a secant line is drawn to the circle 

 ar* -f y 8 = r 2 ; find the locus of the middle point of the chord \\hii li tli. 

 circle cuts from this secant line, as the secant revolves about the given 

 fixed point (r,, yj. 



17. Prove analytically that an angle inscribed in a semicircle is a 

 right angle. 



18. Prove analytically that a radius drawn perpendicular to a chord 

 of a circle bisects that chord. 



19. Show that the distances of two points from the center of a circle 

 are proportional to the distances of each from the polar of the other. 



20. Two straight lines touch the circle x 9 + y 2 5 x 3 y + C 

 one at the point (1, 1) and the other at the point (2, 3); find the polo 

 of the chord of contact of these tangents. 



21. Find he condition among the coefficients that must be satisfied 

 if the circles 



** -f y a + 2 Op + 2 F,y = and ** -f y + 2 Gp + 2/^ = 

 shall touch each other at the origin. 



22. Determine F and C so that the circle 



z* + y a + 20 x -f 2 Fy + C = 

 shall cut eaeh of the circles 



ar + y t -4a:-2y^4 = and 

 at right angles (cf. Art. 1 



