An Elementary Treatise On Conic Sections And Algebraic Geometry With Numerous
An Elementary Treatise On Conic Sections And Algebraic Geometry With Numerous
G Hale George Hale Puckle
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121; then, if we suppose (x y ) to be on the circle, so that x 2 + 7/ 2 = r 3, the equation becomes ?/ 2 M 2 + 2x y m + x * = 0, or my + x = ; hence /j. And fj. Are each equal to -, , and equations (2) of Art. 121 become since the tangents which can be drawn from (x y ) now coincide. 121. To determine the equations to the tangents drawn to a circle from any point (x y). Let the equation to the tangent be y mx = + rjl + m 2 ; THE TANGENT. 125 then, since it passes through (xy), the co-ordinates ...of that point satisfy the equation, and we have (if - mx) z = r 2 (1 + ra 2 ), or (x* - r 2 ) m* - Zxy m + y z - r 2 = (1), a quadratic to determine the two values of m, in the equa tions to the two tangents which can be drawn to the circle from (xy). If p and /* be the two roots, the required equations will be (Art. 29, Cor. 1) y-y = n(x-x), y-y =^ (x-x) (2). COR. Solving equation (1), we have y r Jx* + y - r 2 whence we see that the values of ra are real, if # 2 + y* > ?- 2, or the point is outside the circle; they are equal, when x l + ?/ 2 = r 2, or the point is on the circle ; and they are imaginary, when x* -f- y" 2
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