CONIC SECTIONS, ANALYTICAL. !*>!> 



and this may be adapted to all the special cases. The equation of 

 the tangent at a point (A", Y) to the parabola 



s 



the signs of the radicals in the equation of the tangent being 

 determined by the corresponding signs in the equation of the 

 curve at the point ( X, Y). The equation of the polar of course 

 cannot be expressed in this form. 



The equation of two tangents from (X t Y) is 

 (a.Y f + bY a + c + 2aY+ 2b'X + 2c'XY) 

 (ax 9 + bif + c + 2a'y + 2b f x + 2c'xy) 



G7G. The equation 



Ix (bx + cy) - my (ax + by) = 



represents a pair of conjugate diameters of the conic 

 ax 9 + 2bxy + cy* = d. 



G77. If X ,, + y it = 1 be an ellipse referred to conjugate 



meters inclined at an angle <o, the condition that the 

 x 9 + 2xy cos CD -f y* = r* may touch the ellip-e is 



1\ C08 f ft> 





Hence determine the relations between any conjugate diameters 

 and the axes. 



G78. The axes of the conic oaf + 2&ry + cy" = d make 

 the lines bisecting the angles between the axes of co-ordinates 

 angles 6} prove that 



(a-c)Sntt 



)oosu 



:ig the angle between the axes* 



