TANGENT AND NORMAL. EXAMPLES. 289 



] 0. Find at what angle the curve y* = 2ax cuts the curve 

 x 3 - 3axy + y 3 = 0. 



Results. The curves meet at the origin ; here the 

 first curve has the axis of y for its tangent, and the 

 second curve has both the axes for tangents. The 

 curves also meet at the point x = a 1/2, y = a ^/4 ; and 

 here they meet- at an angle whose cotangent is ^/4. 



#2 ty 2 



11. Tangents are drawn to the ellipse -2+f 2 =l, and the 



Cv Cf 



circle x* + y* a z = Q, at the points where a common 

 ordinate cuts them: shew that if c be the greatest 

 inclination of these tangents 



, a-b 

 tan < = . ,. . 

 2 y\ao) 



12. If tangents be drawn from a point (h, K) to the curve 



fx\ 3 /y\ 3 



whose equation is (-1 +(r) = 1, an ellipse whose 

 \aj \bj 



serniaxes are air] , and 6 [ T j will pass through the 

 \n/ \kj 



points of contact. 



/ x\ 



13. Shew that all the points of the curve y 2 = 4a ( x + a sin - ] 



at which the tangent is parallel to the axis of x lie on 

 a certain parabola. 



14. The normal to a parabola at any point P is produced 



to meet the directrix at Q, and the tangent at P meets 

 the directrix at R : find (1) when QR is a minimum, 

 (2) when the triangle PQR is a minimum. 



Results. (l)o: = -, (2) x = - ; where y* = iaoc is 



o O 



the equation to the parabola. 



T. D. c. 



