. Algebraic geometry; a new treatise on analytical conic sections . a (a; + aij). This passes through Q; .-. y^y^ = 2a{x^ + X2) (1) But yy2 = 2a(x + X2) is the polar of Q, and by (1) this straightline passes through [x-^, y-^ P, which proves the proposition. 145. To fimd the locus of the middle pmmis of a system of parallelchords. Let GQ be a chord of the system, making an angle d with theaxis of X, 6 being constant. Let (ajj, ^j) be the co-ordinates of V, its middle point. The equation of the chordmay be written cos d sin 6x = x-^ + r cos 0and y = y-^+ram6;.. where the chord meets thecurve, w
. Algebraic geometry; a new treatise on analytical conic sections . a (a; + aij). This passes through Q; .-. y^y^ = 2a{x^ + X2) (1) But yy2 = 2a(x + X2) is the polar of Q, and by (1) this straightline passes through [x-^, y-^ P, which proves the proposition. 145. To fimd the locus of the middle pmmis of a system of parallelchords. Let GQ be a chord of the system, making an angle d with theaxis of X, 6 being constant. Let (ajj, ^j) be the co-ordinates of V, its middle point. The equation of the chordmay be written cos d sin 6x = x-^ + r cos 0and y = y-^+ram6;.. where the chord meets thecurve, we have by substitution, (2/1 + r sin 6)^ = 4:a{x-i + r cos 6) or r^sin^^ + 2r (^1 sin ^ - 2a cos 6) + y-^ - 4aa;i = 0. The chord meets the curve at Q and Q, and VQ, VQ, the roots of this quadratic, are equal but of opposite sign; .. ^1 sin ^ - 2a cos 6 = 0. But (sj, ^j) is any point on the locus; .. suppressing the sufifix, = 2a cos Q is the equation of the locus of V. 2aThis may be written y = -—2> a-^d represents a straight line parallel to the Fig. 90. ART. 147.] THE PARABOLA. 133 Def. The straight line which bisects a series of parallel chordsis called the diameter of those chords, and the semi-chords arecalled the ordinates of that diameter. 146. If the diameter PV meets the curve at P, the ordinate of P = the ordinate of V = -—^ = —, where m is the slope of thechord. *^°^ ™ .. the abscissa of P = —j, and y = mx-\— is the tangent at P. Therefore the tangent at the end of a diameter is parallel tothe chords bisected by that diameter. This may also be seen by letting the chord QQ move parallelto itself until V coincides with P. The equal portions VQ, VQ vanish together when V coincideswith P, and the chord becomes a tangent. 147. To find the equation of a chord of the parabola ■f = iax interms of the co-ordinates (x-^, y^ „of its middle point. Let QQ be the chord whosemiddle point is V(a;,, y,), andlet it make an angle 9 withthe axis of x. It
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