BOOK OF MATHEMATICAL PROBLEMS. 



974. Two circles of radii 6, a b, respectively roll within a 

 circle of radius a, their points of contact with the fixed circle 

 being originally coincident, and the circles rolling in opposite 

 directions in such a manner that the velocities of points on the 

 circles with respect to their respective centres are equal : prove 

 that they will always intersect in the point which was originally 

 the point of contact. 



975. In a hypocycloid the radius of the rolling circle is 

 ^- - - times the radius of the fixed circle (n integral) ; prove 



that the locus of the point of intersection of perpendicular tan- 

 gents is a circle ; and that the line joining the points of contact 

 is also a tangent to a hypocycloid having the same fixed circle. 



976. A circle is drawn to touch a cardioid and pass through 

 the cusp : prove that the locus of its centre is a circle. If two such 

 circles be drawn, and through their second point of intersection 

 any straight line be drawn, the tangents to the circles at the ends 

 of this straight line will intersect on the cardioid. 



977. If S, II be foci of a lemniscate, PT the tangent at any 

 point P, 



SP-SffP 



and if 0, <f> be the acute angles which the tangent makes with the 

 focal distances, 



978. Two circles touch the curve r m = a m cos mO in the points 

 P, Q and touch each other in the pole S : prove that the angle 



J *SQ is equal to , n being a positive or negative integer. 



979. The locus of the centre of a circle touching the curve 

 r m = a m cos mO and passing through the pole is the curve 

 (2r) n = a" cos nO, {n(l-m) = m}. 



