Section A5Conic Sections599Now, the graph of Equation 11 is a (real or degenerate)(a)parabolaif Aor C0; that is, if A C0;(b)ellipseif Aand Chave the same sign; that is, if A C0;(c)hyperbolaif Aand Chave opposite signs; that is, if A C0.It can also be verified from Equations 6 that for any rotation of axes,B24ACB24A C.(12)This means that the quantity B24ACis not changed by a rotation. But when we rotatethrough the angle agiven by Equation 10,Bbecomes zero, soB24AC4A C.Since the curve is a parabola if A C0, an ellipse if A C0, and a hyperbola if A C0,the curve must be a parabola if B24AC0, an ellipse if B24AC0, and a hyperbolaif B24AC0. The number B24ACis called the discriminant of Equation 9.EXAMPLE 3Applying the Discriminant Test(a) 3x26xy3y22x70 represents a parabola becauseB24AC624•3•336360.(b) x2xyy210 represents an ellipse becauseB24AC124•1•130.(c) xyy25y10 represents a hyperbola becauseB24AC124 0110.Now try Exercise 15.Technology ApplicationHow Some Calculators Use Rotations to Evaluate Sines and CosinesSome calculators use rotations to calculate sines and cosines of arbitrary angles. The pro-cedure goes something like this: The calculator has, stored,1.ten angles or so, saya1sin1101,a2sin1102,… ,a10sin11010,and2.twenty numbers, the sines and cosines of the angles a1,a2, … ,a10.To calculate the sine and cosine of an arbitrary angle u, we enter u(in radians) into thecalculator. The calculator substracts or adds multiples of 2pto uto replace uby the anglebetween 0 and 2pthat has the same sine and cosine as u(we continue to call the angle u).Discriminant TestWith the understanding that occasional degenerate cases may arise, the quadraticcurve Ax2BxyCy2DxEyF0 is(a)aparabolaif B24AC0,(b)an ellipseif B24AC0,(c)a hyperbolaif B24AC0.
600AppendicesTable A5.3Examples of quadratic curvesAx2BxyCy2DxEyF0ABCDEFEquationRemarksCircle114x2y24AC;F0Parabola19y29xQuadratic in y,linear in xEllipse49364x29y236A,Chave same sign,AC;F0Hyperbola111x2y21A,Chave opposite signsOne line (still 1x20y-axisa conic section)Intersecting lines Factors to (still a conic 1111xyxy10x1y10,section)so x1,y1Parallel lines Factors to (not a conic 132x23x20x1x20,section)so x1,x2Point11x2y20The originNo graph11x21No graphThe calculator then “writes”uas a sum of multiples of a1(as many as possible withoutovershooting) plus multiples of a2(again, as many as possible), and so on, working its wayto a10. This givesum1a1m2a2…m10a10.