Elements of analytical geometry and the differential and integral calculus . quation of the curve, (y^=2px) for thepoint F, we must necessarily make x=^p, thenthe equation becomes y=p. That is, FF=FI>=2 VF, or FB=4: VF. Q. E. D. Corollary. It will be observed that CF and DBsire squares,and the line DF or its equal FF is the quantity represented byJ). It is the same for the same parabola, but different in differ-ent parabolas. PROPOSITION W. To find the equation of a tangent line to the parabola. Let the line SPQ cut the parabolain two points P and Q. Let ir, y, be the general co-ordi-nates
Elements of analytical geometry and the differential and integral calculus . quation of the curve, (y^=2px) for thepoint F, we must necessarily make x=^p, thenthe equation becomes y=p. That is, FF=FI>=2 VF, or FB=4: VF. Q. E. D. Corollary. It will be observed that CF and DBsire squares,and the line DF or its equal FF is the quantity represented byJ). It is the same for the same parabola, but different in differ-ent parabolas. PROPOSITION W. To find the equation of a tangent line to the parabola. Let the line SPQ cut the parabolain two points P and Q. Let ir, y, be the general co-ordi-nates of any point in the line as S;x\ y the co-ordinates of the point F;and x, y, the co-ordinates of thepoint Q. The equation of a straight line which passes through the twopoints, S and P, must be of the form y—y^a{x—x), (1) We require the value of a when SP is tangent to the curve. If the same line passes through the two points S and Q, wemust have y—y=a(x—x). (2) And the same line passing through the two points P and Q willrequire the equation y—y—a{x—x). (3). 68 ANALYTICAL GEOMETRY. The two points P and Q being in the curve, also requirey^~9,px\ (4) And y^=2px\ (5) By subtraction y^—y^=^9,p(x—x). Or {y—y)(y+y)^^p{x—x) (6)Dividing (6) by (3) will give 2/+y= ^p (7) Now conceive the line >S^ to turn on the point P as a centeruntil Qflows* into P, then we shall have Put this value oiy in (7), and we find y (8) This value of a put in (1) will reduce that equation toyy—y^=px--px\ But y^=2px By addition yy=p>{x-\X) and this is the equation sought, x, y, are the co-ordinates of anypoint in the line, and x\ y\ the co-ordinates of the tangent pointin the curve. Corollary. To find the point inwhich the tangent meets the axis ofX, we must make 2/=0, this makes p{X\-x)=^0. Or x=—X. That is, VD=^ VT, or the sub-tangent is bisected by the vertex. Hence, to draw a tangent line from any given point, as P, wedraw the ordinate FD, then make TV= VD, and from the poi
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