Elements of analytical geometry and the differential and integral calculus . erred toco-ordinates requires equations of the second degree. PROPOSITION find the equation of the circle. Let the origin be the center ofthe circle. Draw AM\.o any pointin the circumference, and let fallMP perpendicular to the axis of AP=x, PM=:y and AM=P. Then the right angled triangle^PJf gives x^+y=P^ (1) and this is the equation of the circlewhen the zero point is the center. When y=:0, x^ = R^, or ±:x=P, that is, Pis at X or A\Whenar=0, y=R^, or ±y=R, showing that if on the cir-cumference is then at Y
Elements of analytical geometry and the differential and integral calculus . erred toco-ordinates requires equations of the second degree. PROPOSITION find the equation of the circle. Let the origin be the center ofthe circle. Draw AM\.o any pointin the circumference, and let fallMP perpendicular to the axis of AP=x, PM=:y and AM=P. Then the right angled triangle^PJf gives x^+y=P^ (1) and this is the equation of the circlewhen the zero point is the center. When y=:0, x^ = R^, or ±:x=P, that is, Pis at X or A\Whenar=0, y=R^, or ±y=R, showing that if on the cir-cumference is then at Yov Y\ When X is positive, then P is on the right of the axis of Y,and when negative, P is on the left of that axis, or between Aand A\ When we make radius unity, as we often do in trigonometry,then x^-\-y^ = \, and then giving to x oy y any value plus orminus within the limit of unity, the equation will give us thecorresponding value of the other letter. In trigonometry y is called the sine of the arc XM, and x itscosine. Hence in trigonometry we have |-cos.^ = 2a ANALYTICAL GEOMETRY. Now if we remove the origin to A and call the distanceAF=x, then AF=x—E, and the triangle AFH gives Whence y^=2JRx—x^. (2) This is the equation of the circle, when the origin is on thecircumference. When x=0 y=0 at the same time. When x is greater than2M, y becomes imaginary, showing that such an hypothesis is in-consistent with the existence of the circle. There is still a more general equation of the circle when thezero point is neither at the center nor in the figure in the margin will fully illustrate. Let AB=c,^ BC==b. PutAP=x, or AP=zj:, and FMovFM==y, CM, CM, &c. each In the circle we observe fourequal right angled numerical expression is thesame for each. Signs only indi-cate positions. Now in case CDM is the tri-angle wefx upon,We put AP=x, then BF=CD={x—c),FM=y, MI)=y—CB=(y—h].
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Keywords: ., bookauthorrobinson, bookcentury1800, bookdecade1850, bookyear1856
