Plane and solid analytic geometry; an elementary textbook . plest. 80 ANALYTIC GEOMETRY [Ch. VII, § 55 According to the definition we must first write theequation of a secant through two points, and then findthe limiting form which this equation approaches whenthe two points approach coincidence. Let (xv y{) and (x1 + \ y^ + &) be the coordinatesof Px and P2, adjacent points on the circle x2 + y2 = equation of the line through these two points is (by [5]) y — y\ = k. x — xx h If we let P2 approach Pv h and k will approach zero,and the limit of the second member will be indeterminate. Th
Plane and solid analytic geometry; an elementary textbook . plest. 80 ANALYTIC GEOMETRY [Ch. VII, § 55 According to the definition we must first write theequation of a secant through two points, and then findthe limiting form which this equation approaches whenthe two points approach coincidence. Let (xv y{) and (x1 + \ y^ + &) be the coordinatesof Px and P2, adjacent points on the circle x2 + y2 = equation of the line through these two points is (by [5]) y — y\ = k. x — xx h If we let P2 approach Pv h and k will approach zero,and the limit of the second member will be indeterminate. This would be neces-sary since we havemade no use of thefact that P2 must ap-proach Px along thecircle. Unless P2 ap-proaches Px alongsome curve, PXP2 willhave no limiting posi-tion. It will there-fore be necessary todetermine in the case of each curve the value of the kexpression -? In the case of the circle about the origin,ri the coordinates of the points Px and P2 must satisfy the equation x2 + y2 = r2. We have, therefore, (1) rrx2 + y^ = r2,. Fig. 47. and (2) x2 + 2hx1 + h2 + y2 + 2 kyl + k2 = r2. Ch. VII, § 50J THE CIRCLE 81 Subtracting (1) from (2), we have 2 hxx + A2 + 2 kyt + k2 = 0, kor, transposing and solving for -, ill h_ 2 xx + ht h~ 2yi + k Substituting in the former equation of the secant PXPVwe see that y _ ,fi _ 2x^ + h x — xx 2y1-\-k is another form of its equation in the circle x2 -f y2 = now we let h and k decrease, the limit of the second member is no longer indeterminate, but becomes K The equation of the tangent is therefore * V-Vx= \x - xx yl which by the aid of (1) reduces to scias + y\V = r2. [28] Let the student show by the same method that the equation of the tangent to the circle a* + yZ + Dx + IJy + F=0 is xix + 2/12/ +?( + ^) + f (V + 2/i)+ F = 0. [29] 56. Normal. — The normal at any point of a curve isthe line through the point, perpendicular to the tangent atthe point. Its equation can be obtained by first writingthe equation of t
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