103 



CONIC SECTIONS, ANALYTICAL. 



CARTESIAN CO-ORDINATES. 



I. Straight Line, Linear Transformation, Circle. 



\ v any question relating to the intersections of a curve and 

 two straight lines, it is generally convenient to use one equation 



>enting both straight lines. Thus, to prove the theorem 

 that " Any chord of a given conic subtending a right angle at a 



< point of the conic passes through a fixed ]K>int in the nor- 

 mal at the given point;" we may take the equation of the conic 



red to the tangent and normal at a point 



the equation of any pair of straight lines through this point, at 

 angles to each other, is 



and at the points of intersection 



(a + c) v? + 2 ( 6 4- cX) xy + 2x = ; 

 or, at the points other than the origin, 



(a + c)x + 2 (6 + c\)y 4- 2 0, 



is therefore the equation of a chord subtending a right 

 at the origin. Thi- J.IISS.H through the point y 0, 

 (a + r) a: + 2 = ; a fixed point in the normal. 



< be given as the intersections of a straight line and 



ic, the equation of the line joining those pointe to the origin 



may be found immediately, d ust be a homogeneous equa- 



