CONIC SECTIONS, ANALYTICAL. 



that the circles of either system have a common radical 

 axis which is the line of centres of the circles of the other 

 m. 



I. Given two circles, a tangent to one at P meets the 

 polar of P with respect to the other in P'; prove that the circle 

 on PP' as diameter will pass through two fixed points, which will 

 be imaginary or real, according as the circles intersect in real or 

 imaginary points. 



545. One circle lies entirely within another, a tangent to the 

 inner meets the outer in P, P', and the radical axis in Q: if be 

 the internal vanishing circle which has the same radical axis, the 



.. . PSP' SQP . 

 ratio sin ^ : cos - - is constant. 



a J 



546. Prove that the equation 



{x cos (a + ft) + y sin (a + ft) - a cos (a - /)} 



{x cos (y + 8) + y sin (y + 8) - a cos (y - 8)} 



= {x cos (a + y) -f y sin (a -I- y) - a 008 (a - y)} 



{x COB (ft + 8) + y sin (ft + 8) - a cos {ft - 5)} 



is equivalent to the equation of + y* = a* ; and state the property 

 of tin- circle expressed by the equation in this form. 



7. The radii of two circles are R, r, the distance between 

 their centres is J(J^+2r') t and r<l'A' ) prove that an infinite 

 ..;' triangles can be inscribed in tin- first, which are self- 

 with respect to the second : and that an infinite number 

 of triangles can be circumscribed to the second which are self- 

 i gate with respect to the first 



548. A triangle is inscribed in tin- ' + y* Jf and two 



of its sides touch the circle (x - 5)' + y f = r* ; prove that the third 

 side will touch the circle 



> circle coincides with the second if 8* I? * - 



