PRINCIPAL DIAMETER AND ITS TANGENT. CHORDS. 325 
2. AP is the arc of a conic section, of which the vertex is A; 
PG the normal, and PK a perpendicular to the chord AP, meet 
the axis in G and K. To shew that GK is equal to half the 
latus-rectum. 
3. To find the locus of the middle point of the portion of the 
normal to a conic section which is included between the curve 
and the axis. 
The equation to the conic section being 
y* = mx + mx 2 , 
that to the required locus will be 
(n + 2) 2 y 2 — mx* — mx + \ (m + l) = 0. 
Lardner: Algebraic Geometry, p. 149. 
Section II. 
Referred to a Principal Diameter and its Tangent. Chords. 
1. From one extremity A of a principal axis of a conic 
section, a given straight line AP is drawn to cut the curve in P; 
to find the equation to the line joining P to the other extremity 
B of the axis. 
Let the axis AB and the tangent at A be taken as axes of x 
and y respectively. Then the equation to AP will be of the 
f° rm y = ax (l) ? 
and the equation to the conic section of the form 
y 2 = mx + wx 2 (2). 
The equation to AB is y = 0 (3). 
The two lines (1) and (3) may be represented simultaneously by 
the equation y ^ ^ = 0? 
or y* = axy (4). 
Hence, at A, P, P, the intersections of AP, AB, with the curve, 
axy = mx + mx 2 , 
whence dividing by x, we obtain for the equation to BP, which 
passes through P, B, 
ay — m + nx.