. Plane and solid analytic geometry; an elementary textbook. als can bedrawn to (a) an ellipse, (b) a parabola ? 13. Obtain the equation of the tangent at the point P1 ofthe parabola y2 = 2 mx, by determining I in the slope form ofthe equation of the tangent in terms of x^ and yv 78. Theorems concerning tangents and normals. — 1. The tangent and normal at any point of an ellipse bisect theexterior and interior angles respectively between the focalradii drawn to the point of contact. Let PXT and PXN be the tangent and normal to the ellipse -— + &- = 1 at the point Pv We wish to show thata1 bl P
. Plane and solid analytic geometry; an elementary textbook. als can bedrawn to (a) an ellipse, (b) a parabola ? 13. Obtain the equation of the tangent at the point P1 ofthe parabola y2 = 2 mx, by determining I in the slope form ofthe equation of the tangent in terms of x^ and yv 78. Theorems concerning tangents and normals. — 1. The tangent and normal at any point of an ellipse bisect theexterior and interior angles respectively between the focalradii drawn to the point of contact. Let PXT and PXN be the tangent and normal to the ellipse -— + &- = 1 at the point Pv We wish to show thata1 bl PXT bisects the angle FPXK, and that PXN bisects theangle FP^F. It is a well-known theorem of elementary geometry thatthe bisector of an interior angle of a triangle divides theopposite side into segments which are proportional to the 134 ANALYTIC GEOMETRY [Ch. X, § 78 adjacent sides of the triangle. The converse theorem isalso true. It is therefore sufficient to show that FN: NF FF,FPX The equation of the normal PXN is a2yxx - b2xxy = (a2 - b2)x^v. Fig. 71. . a2 —b2Its intercept ON on the X-axis is — xv or, since in a? — b2 a the ellipse — = e2, a1 CN=e2xvAlso, FfC=CF=ae. Hence FN= FC+CN=ae + e2xx = e (a + ex{),and NF=CF - CN = ae - e2xx = e(a - exj. But (by [45]) FF1 = a + g^, and FPX = a — e#r Ch. X, § 78] TANGENTS 135 FP FNHence l = , and the normal bisects the angle FP^F. Since the tangent is perpendicular to the normal,it bisects the supplementary angle FPXK. Note. — It is upon this principle that whispering galleries are con-structed. If the whole or part of the sides of a room is a surface formedby revolving an ellipse about its major axis, all waves of sound, light, orheat starting from one focus and striking this surface will be reflected tothe other focus. 2. In an hyperbola the tangent and normal at any pointbisect the interior and exterior angles respectively between thefocal radii. 3. If an ellipse and hyperbola are confocal (or have thesame foci)
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