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LINES OP THE SECOND ORDER. 
2. If A and B be the extremities of the axis major of a 
conic section, T the point where a tangent at a point P in the 
curve, meets AB, Q TR a line perpendicular to AB and meeting 
AP, BP, in Q, R, respectively; to prove that 
QT= RT. 
3. To find the length of the chord of a conic section, denoted 
by the equation 
y 2 = 2 mx + nx 2 , 
the equation to the chord being 
If 2c denote the length of the chord, 
a _ a (a 2 + ft 2 ), (in 2 a + 2 in ft 2 + not. ft 2 ) 
C = (nd 2 - ft 2 ] 2 
Section III. 
Referred to a Principal Diameter and its Tangent. Focal 
Properties. 
1. From the extremity L of the semi-latus-rectum SB of a 
conic section, a chord LA is drawn to the vertex A of the 
diameter through 8. A tangent is drawn at L. A straight 
line MRP is drawn, through any point M in AS, or AS pro 
duced, at right angles to AS, meeting the chord AL in R, and 
the tangent at L in P. To prove that PR is equal to MS. 
Taking AS, produced indefinitely, as the axis of x, and the 
tangent at A as that of y, the equation to the curve will be, 
y 2 = (1 + e) {2emx — (1 — e) x 2 ). 
The equation to the tangent at any point (x, y) is 
VV ( I \ i < 
—^— = \em — 1 — e) x\ x + emx. 
1 + e v ’