Inhaltsangabe
This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1836 Excerpt: ...(a, b) of the circle, (x-Ay + (y-B) = R2. Let the secant--+ Jl = 1 pass through the a" b" points (a, b), (a', b'); then Otherwise. (Fig. 19.) Let A B be a tangent at any point P of the circle whose centre is C, the common ordinates of the tangent being OA, OB; then its equation is JL + _!-= l. O A O B But M A: MP:: Pm: Cm and NB: NP:: Cm: Pm tangent and normal at that point; then v the normal passes through (a, b) at right angles to the tangent. (Ex. 2, p. 49.) a" = a-b.--, b» = b-a.--. a' b' But a' = a + b., b' = 6 + a. a--; a--A b--A.. the equation required is + y = 1;, a--A, 6--B 6-B a-A or it is-I = 1; M b--a B /A/--oB 1 6-B--) a-A ) but the former is the more symmetrical, being the same as that for the tangent, with the exception of the minus sign instead of plus in the common ordinates, and the reciprocal of the differences of the co-ordinates of the given point and centre. Otherwise. (Fig. 19.) If N' N", the normal at P, have the common ordinates O A', O B', then the equation is a" b" Also, at the point of contact, the equations iL + JL = l (2)' a" b" W. and-a'f + y-V) = R... (3) are simultaneous. And we have now only three equations involving the four unknown quantities, x, y, a", b". To obtain another equation we must adopt the condition of contact, viz. that the angle between the required tangent and the radius of the circle passing through the point of contact is a right angle. Now the distances between the points (a, b), (a', b'); and between (a, b), (x, y), are V (a-a'f + (b-b'f and V (-a)2 + (y-6), and since the latter and R are the legs of a right angled A of which the other is the hypothenuse, we have (x-a)+(y-6)'+R8=(a-a')'+(6-6'), (4)Equations (4) and (3) become, by reduction, a + y'-2...
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Reseña del editor
This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1836 Excerpt: ...(a, b) of the circle, (x-Ay + (y-B) = R2. Let the secant--+ Jl = 1 pass through the a" b" points (a, b), (a', b'); then Otherwise. (Fig. 19.) Let A B be a tangent at any point P of the circle whose centre is C, the common ordinates of the tangent being OA, OB; then its equation is JL + _!-= l. O A O B But M A: MP:: Pm: Cm and NB: NP:: Cm: Pm tangent and normal at that point; then v the normal passes through (a, b) at right angles to the tangent. (Ex. 2, p. 49.) a" = a-b.--, b» = b-a.--. a' b' But a' = a + b., b' = 6 + a. a--; a--A b--A.. the equation required is + y = 1;, a--A, 6--B 6-B a-A or it is-I = 1; M b--a B /A/--oB 1 6-B--) a-A ) but the former is the more symmetrical, being the same as that for the tangent, with the exception of the minus sign instead of plus in the common ordinates, and the reciprocal of the differences of the co-ordinates of the given point and centre. Otherwise. (Fig. 19.) If N' N", the normal at P, have the common ordinates O A', O B', then the equation is a" b" Also, at the point of contact, the equations iL + JL = l (2)' a" b" W. and-a'f + y-V) = R... (3) are simultaneous. And we have now only three equations involving the four unknown quantities, x, y, a", b". To obtain another equation we must adopt the condition of contact, viz. that the angle between the required tangent and the radius of the circle passing through the point of contact is a right angle. Now the distances between the points (a, b), (a', b'); and between (a, b), (x, y), are V (a-a'f + (b-b'f and V (-a)2 + (y-6), and since the latter and R are the legs of a right angled A of which the other is the hypothenuse, we have (x-a)+(y-6)'+R8=(a-a')'+(6-6'), (4)Equations (4) and (3) become, by reduction, a + y'-2...
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