. Algebraic geometry; a new treatise on analytical conic sections . » .. it is the equation of QQ. 143. Pole and Polar. Substitute the words conic sectionfor circle in the definition of Art. 94. To find the polar of the point {x^, y^) mth respect to the parabolay^ = iax. Let P be the point (x-^, y-^ and PAB any chord through P; AQ,BQ the tangents at A and B. It is required to find the locus of Q. Let (A, k) be the co-ordiiiates of Q. Then the equation of its chord of contact AB isyk=2,a(x + h). ART. 148.] THE PAKABOLA. 131 But (x,, y^ is on this line; ..2/1*= 2a{xi + h)., Also (a, k) is (my po
. Algebraic geometry; a new treatise on analytical conic sections . » .. it is the equation of QQ. 143. Pole and Polar. Substitute the words conic sectionfor circle in the definition of Art. 94. To find the polar of the point {x^, y^) mth respect to the parabolay^ = iax. Let P be the point (x-^, y-^ and PAB any chord through P; AQ,BQ the tangents at A and B. It is required to find the locus of Q. Let (A, k) be the co-ordiiiates of Q. Then the equation of its chord of contact AB isyk=2,a(x + h). ART. 148.] THE PAKABOLA. 131 But (x,, y^ is on this line; ..2/1*= 2a{xi + h)., Also (a, k) is (my point on the locus;.. the equation of the locus is yjy=2a(xj+x) or y^i = 2a (a;+ a;j), a straight line. When (x^, y^) is outside the parabola, the polar is the sameas the chord of contact of tangents d/ravm from (x^, y^.. Fia. 89. In the figure, QA, QB are tangents at A and B,-QC, QD „ C and D, QE, QF „ E and F. QQQH is the polar of P and PH is a tangent at H. 132 THE PAEABOLA. [chap. VIII. 144. If the polar of the point P passes through the point Q, thepolar of the point Q passes through the point P. Let (ajj, ^j) be the co-ordinates of P, {x^, y<^ those of equation of the polar of P with respect to the parabolay^ = iax is yy-^ = 2a (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
Size: 1453px × 1720px
Photo credit: © Reading Room 2020 / Alamy / Afripics
License: Licensed
Model Released: No
Keywords: ., bookcentury1900, bookdecade1910, bookpublisherlondo, bookyear1916
