CONIC SECTIONS, ANALYTICAL. 177 



properties of the foci, especially of the parabola, may thus bo 

 .-.ilized by projections. Thus, remembering that the directrix 

 is the polar of the focus, we see that if a conic be inscribed in a 

 triangle ABC, and two tangents be drawn dividing BC harmoni- 

 cally, their point of intersection lies on the polar of -4. So also, 

 the locus of the intersection of tangents to a parabola, includ- 

 ing a constant angle, is a conic having the same focus and directrix, 

 it follows that, if a conic be inscribed in a triangle ABC, and two 

 tangents be drawn dividing BC in a constant range, the locus of 

 their point of interaction is a conic having double contact with 

 the former at the points where AB t AC touch it. 



The circular ^points at infinity have singular properties in 

 relation to many other curves. All epicycloids and hypocycloids 

 pass through them, the cardioid has cusps at them, and may be 

 projected into a three-cusped hypocycloid. 



823. If two conies have a common focus S, and two common 

 tangents PQ, FQ[, the angles FSQ, PSQ will be equal or sup- 

 plementary. 



824. If two conies have a common focus and equal minor 

 axes, their common tangents will be parallel 



825. If two conies have a common focus and equal latera recta, 

 one of their common chords will pass through the common focus. 



826. If S be the focus of a conic and A any point, the 

 straight line drawn through >S T at right angles to SA will meet 

 the polar of A on the dire* 



827. If be a fixed point on a conic, J, B, C any three 



points on the conic, and the straight lines through at 



angles to OA, OB, OC meet BC t CA, AB respectively in 



, C' t the straight line A f ffC f will meet the normal at in a 



fixed point 



828. Given a conic and a point ; prove that there are two 

 straight lines, such that the distance between any two points on 

 one of them, conjugate to the conic, subtends a right angle at 0. 



w. 12 



