. Plane and solid analytic geometry . — -VB^ 2A 2A A^O, From these formulas the truth of the following theorem atonce becomes apparent. Theorem 1. TJie roots of the quadratic equation (1) areequal if and only if ^2-4^0=0. The quantity ^- — ^AC is known as the discriminant of thequadratic equation (1). By means of the theorem we shall solve the following prob-lem. Problem. Let it be required to find the equation of thetangent to the parabola (2) f = Qx, which is of slope \. Let i be a line of slope \which meets the parabola intwo points, Pj and Pg. If weallow L to move parallel to ^itself
. Plane and solid analytic geometry . — -VB^ 2A 2A A^O, From these formulas the truth of the following theorem atonce becomes apparent. Theorem 1. TJie roots of the quadratic equation (1) areequal if and only if ^2-4^0=0. The quantity ^- — ^AC is known as the discriminant of thequadratic equation (1). By means of the theorem we shall solve the following prob-lem. Problem. Let it be required to find the equation of thetangent to the parabola (2) f = Qx, which is of slope \. Let i be a line of slope \which meets the parabola intwo points, Pj and Pg. If weallow L to move parallel to ^itself toward the tangent, T,the points P^ and P^ will movealong the curve toward P, thepoint of tangency of T\ and if L approach T as its limit, the points P^ and P2 will approachthe one point P as their limit. It is clear that these considerations are valid for any , we may state the following theorem. Theorem 2. A line which meets a conic intersects it ingeneral in two points. If these two points approach coincidence. Fig. 5 176 ANALYTIC GEOMETRY in a single point, the limiting position of the line is a tangent tothe conic* In applying Theorem 2 to the problem in hand, let us denotethe intercept of the tangent T on the axis of y by ft. Theequation of T is, then, (3) 2/ = i-a^ + )S. The coordinates of the point P, in which T is tangent to theparabola, are obtained by solving equations (2) and (3) simul-taneously. Substituting in (2) the value of y given by (3),we have or (4) a;2 + 4(^-6)aj +4/32 = 0. The roots of equation (4) are equal, since they are both theabscissa of P. Accordingly, by Theorem 1, the discriminantof (4) is zero. Hence 16 (/3 - 6)2- 16^2 ^0, or -^ 12y8 + 36 = 0. Thus /8 = 3, and the tangent to the parabola (2) whose slopeis -i- has the equation (5) x-22/ + 6 = 0. If in (4) we set /? = 3, the resulting equation, a;2-12a; + 36 = 0, has equal roots, as it should. The common value is cc = 6, andthe corresponding value of y, from (2), is 1/ = 6. Th
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