Elements of analytical geometry and the differential and integral calculus . d x and y the co-ordinates of the point P, equation (8) of the last propositiongives us x — -• This value of a put in (1) and we have y—y =—- {x—x)y for the equation sought. This equation combined with that of the circlex^+y^^R^will determine the values of x and y, and as there will be twovalues to each, numerically equal, it shows that two equal tan-gents can be drawn from H, or from any point without the circle,which is obviously true. Scholium. We can find the value of the tangent FT bymeans of the similar triangle
Elements of analytical geometry and the differential and integral calculus . d x and y the co-ordinates of the point P, equation (8) of the last propositiongives us x — -• This value of a put in (1) and we have y—y =—- {x—x)y for the equation sought. This equation combined with that of the circlex^+y^^R^will determine the values of x and y, and as there will be twovalues to each, numerically equal, it shows that two equal tan-gents can be drawn from H, or from any point without the circle,which is obviously true. Scholium. We can find the value of the tangent FT bymeans of the similar triangles ABP, PBT, which giveX \ R \ \ y \ FT. ft=rI. X More general and elegant formulas will be found in the cal-culus for the normals, subnormals^ tangents, and suhtangentsapplicable to all the conic sections. Note to Propositions III and IV of this Chapter.—In the investiga-tion of these propositions we followed in the footsteps of others, only hopingto be more definite and clear. But were we only in pursuit of results, wewould have been more brief and THE CIRCLE. 35 In these propositions it is not assumed that the radius of the circle is atright angles to its tangent when drawn from the center to the point of con-tact, but we see no propriety in excluding this geometrical truth so wellknown in elementary geometry, especially when we consider that we haveall along used the symbol a to represent the tangent of angles on the admis-sion that the tangent of an angle was a line drawn at right angles to ihrradius from the extremity of the radius. Using this truth we would not draw a linecutting the curve in two points, but woulddraw the tangent line PT at once, and adinitthat the angle APT was a right angle. Thenit is clear that the angle APB= the anglePTB. Now to find the equation of the line, we letx and y represent the co-ordinates of the pointP, and X and y the general co-ordinates of the line, and a the tangent of its angle with the axis of X, then by (Prop.
Size: 1769px × 1413px
Photo credit: © The Reading Room / Alamy / Afripics
License: Licensed
Model Released: No
Keywords: ., bookauthorrobinson, bookcentury1800, bookdecade1850, bookyear1856
