Plane and solid analytic geometry; an elementary textbook . Fig. 73. to OX. We wish to prove that the angles TPXF andFTPX are equal. If we let the abscissa of Px be xv its ordinate will be± V2 mxv since Px is a point on the parabola. Then theequation of the tangent at Px is ± ^/2mx1 • y = m (x + x^). If in this equation we let y = 0, we find the interceptOT to be —x,.Hence TO m xv and TF=x1 + ^. But FPX = \ (xx - |Y + 2 mx1 = xx + Cii. X, § 78] TANGENTS 137 Hence TF=FPV and the angles TPXF and FTP1 are equal. What other angles are also equal in the figure ? Note. — Parabolic reflectors depend
Plane and solid analytic geometry; an elementary textbook . Fig. 73. to OX. We wish to prove that the angles TPXF andFTPX are equal. If we let the abscissa of Px be xv its ordinate will be± V2 mxv since Px is a point on the parabola. Then theequation of the tangent at Px is ± ^/2mx1 • y = m (x + x^). If in this equation we let y = 0, we find the interceptOT to be —x,.Hence TO m xv and TF=x1 + ^. But FPX = \ (xx - |Y + 2 mx1 = xx + Cii. X, § 78] TANGENTS 137 Hence TF=FPV and the angles TPXF and FTP1 are equal. What other angles are also equal in the figure ? Note. — Parabolic reflectors depend on this principle. If a surface isformed by revolving a parabola about its axis, all waves of light, etc.,which start from the focus will be reflected in lines parallel to the axis ofthe parabola. 5. Two parabolas which have the same focus and axis,but which are turned in opposite directions, cut each otherorthogonally. 6. The chord of contact of tangents to a parabola fromany point on the directrix passes through the Fig. 74. The coordinates of any point L on the directrix maybe represented by ( — —, yA. The equation of PXPV thechord of contact of tangents from this point, is yxy = mx m? y = lx + m21 X ml y = ~T ~ ~2 138 ANALYTIC GEOMETRY [Ch. X, § 78 Since the coordinates of the focus (—, 0 J satisfy this equation, the chord of contact must pass through thefocus. Let the student prove the converse theorem, viz.: Tan-gents at the ends of any focal chord meet on the directrix. 7. Prove that the same theorems hold for the ellipseand hyperbola. 8. Any two -perpendicular tangents to the parabola meeton the directrix. Two perpendicular tangents may be represented by theequations and By solving these equations simultaneously, the point ofintersection of the two tangents is found to be r_m m(\ - P)-| which is evidently a point on the directrix. Let the student prove the converse theorem, viz.: Twotangents to a parabola from any point on the directrix areperpen
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