. Algebraic geometry; a new treatise on analytical conic sections . t this line touches the curve at the point (xj, y^). Then yyi = 2a(x + x-^) is the equation of the tangent. .. this equation, 2ax-yy^ + 2aXj^ = 0 must be identicalwith lx + my + n = 0, for the two equations represent the samestraight line. .. comparing coefficients : ~= -i^ = -^ ■ I m n , lam , n whence y-y— y- and x-^ = -j. But ^1^ = 4aa;i, for (Kj, y^) is on the curve; . ia^m^ ian •■ W^Tor am^ = In is the required might have found this condition in the following manner ;From the equation y^ = iax, substitut
. Algebraic geometry; a new treatise on analytical conic sections . t this line touches the curve at the point (xj, y^). Then yyi = 2a(x + x-^) is the equation of the tangent. .. this equation, 2ax-yy^ + 2aXj^ = 0 must be identicalwith lx + my + n = 0, for the two equations represent the samestraight line. .. comparing coefficients : ~= -i^ = -^ ■ I m n , lam , n whence y-y— y- and x-^ = -j. But ^1^ = 4aa;i, for (Kj, y^) is on the curve; . ia^m^ ian •■ W^Tor am^ = In is the required might have found this condition in the following manner ;From the equation y^ = iax, substitute for x in the equationh + my + n = 0,and we have at the points where the line meets the curve -^ + my+n = 0 or ly^ + Aamy + ian = 0. ART. 142.] THE PARABOLA. 129 But if the line touches the parabola, this quadratic has equalroots. /. 16a%2=16am«; (J2 = 4ac) .. am^ = In is the required condition, as before. 141. To find the equation of the normal to the parabola y^ = 4:ax atthe point (ajj, y^). The equation of the tangent at (a^, y^) is yyi = 2a(x + x^).. 2a Fio. 87. Its slope = - .. the slope of the normal, which is perpendicular to thetangent, is -P. {mm = - 1) Also, the normal passes through the point (ajy yj); .•. its equation is y-y^= -^{x-Xj). [y-y^ =m(x-xj] 142. Tangents are drawn from the point (x^, y^ to the parabolay^ = 4aa! / to find the equation of their chord of R be the point {x-^, y-^; RQ, RQ the I 130 THE PARABOLA. [chap. VIII. It is required to find the equation of (h-i, kj) be the co-ordinates of Q, (Aji ^2) *^^ co-ordinatesof Q. The equation of RQ, the tangent at Q, is yk^ = 2a{x + hj).RQ „ Q,is yk2 = 2a{x + h2).
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