. Algebraic geometry; a new treatise on analytical conic sections . V is the middle point of PQ : find the co-ordinates of thenoint R where the diameter through V meets the curve. 6. If the normal y=mx - 2am - am^ meets the curve y^=iax at P and Q,find the co-ordinates of the middle point of PQ. Deduce the equation of the locus of the middle points of a series ofnormal chords. 7. Prove that the chord y-x \/2 -I- 4a ij2=0 is a normal to the parabolay^=4aa;, and that its length is 6v3. a. 8. If the sum of the slopes of two normals to the parabola y^=iax isconstant ( = k), find the locus of their
. Algebraic geometry; a new treatise on analytical conic sections . V is the middle point of PQ : find the co-ordinates of thenoint R where the diameter through V meets the curve. 6. If the normal y=mx - 2am - am^ meets the curve y^=iax at P and Q,find the co-ordinates of the middle point of PQ. Deduce the equation of the locus of the middle points of a series ofnormal chords. 7. Prove that the chord y-x \/2 -I- 4a ij2=0 is a normal to the parabolay^=4aa;, and that its length is 6v3. a. 8. If the sum of the slopes of two normals to the parabola y^=iax isconstant ( = k), find the locus of their intersection. ART. 180.] PROPERTIES OF THE PARABOLA. 147 9. Pairs of normals to the parabola y^=4oa! make complementaryangles with the axis ; find the loous of their intersection. 10. Prove that the normal y=mx-2am-arn^ to the parabola y^=iaxmeets the perpendicular normal at a point whose abscissa is a(m= + l + ±,). PROPERTIES OF THE PARABOLA. 159. If the tangent at P, a point on a parabola, meets the axis at T,amd PN is the ordinate at P, AN = AT, and SP = Let (aij, yj) be the co-ordinates of the equation of the tangent PT is yy.^~2a(x + Xj).At T, a point on this line, y = Q;.. 2a(x + x-^) = 0; .. x= -x^ (for a is not equal to zero),or AT = AN, but is of opposite opposite sign shows that AN and AT are drawn in oppositedirections. Also SP=PK = NX = AX-I-AN = AS + AT = ST. Def. NT is called the subtangent. Hence the subtangent is bisected at the vertex. 160. The tangent at P, a point on a parabola, bisects the anglebetween PK the perpendicular on the directrix, and SP the focal z. KPT = ^ PTS, by parallels, = , for ST = SP. 148 PROPERTIES OF THE PARABOLA, [chap, viii 161. If the perpendicular from the focus upon the tangent at Pmeets it at Y, the point Y lies on the tangent at the vertex. This is proved in Art. 152. 162. If SY is dravm frrni the focus S at right angles to the tangentatP, SY2 =
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