. Algebraic geometry; a new treatise on analytical conic sections . radical axis. Def. Such a system of circles is said to be co-axal. 113. Example i. Find the equation of a system of circles which aZl havethe straight line 3a;-5y=7/or their radical axis, one circle of the systemhaving its centre at the origin and radius i. x + y^-16 + \(3x-5y-7) = 0 IB the required equation, where \ may haveany value; for it represents a circle passing through the common pointsof the circle x + y-16=0, and the straight line Sx-5y-T = 0. Alsox + y-l6=0 is one of the circles of the system, for this is what thee
. Algebraic geometry; a new treatise on analytical conic sections . radical axis. Def. Such a system of circles is said to be co-axal. 113. Example i. Find the equation of a system of circles which aZl havethe straight line 3a;-5y=7/or their radical axis, one circle of the systemhaving its centre at the origin and radius i. x + y^-16 + \(3x-5y-7) = 0 IB the required equation, where \ may haveany value; for it represents a circle passing through the common pointsof the circle x + y-16=0, and the straight line Sx-5y-T = 0. Alsox + y-l6=0 is one of the circles of the system, for this is what theequation of the system becomes when we take X equal to zero. Example ii Find the equation of the circle which passes through thepoints of intersection of x + y^ + ix-Sy + i=(i and x^ + y^+2x+2y-2=0,and aXso through the origin. The general equation of a circle through the intersections of the givencircles is x+y^ + iix-»y + 4: + \{x^ + y^ + 2x + Qy-2)=0 (1) But the required circle passes through the origin (0, 0). .•. the values (0, 0) must satisfy equation (1).. Exs. VII. b.] RADICAL AXIS OF TWO CIRCLES, 109 .-. 4-2X^0 and X=2. Substituting this value of \ in equation (1), we have for the reauiredequation „ „ „ Ezamples VII. b. 1. Find the equation of the radical axis of the circles a;2+y2=l, x^ + y^-4x-6y+n= a figure on squared paper, unit one inch. 2. Prove that the radical axis of two circles bisects their commontangents. 3. Find the ra,dical axis of the circles x^ + y^-2x-2y + l=0 and 2x + 2y^-ix-5y-i=0. 4. Find the radical axis of the circles x + y+2ax-c^ = 0 and x^ + y^ + 2bx-c=0, •and prove that the distance of any point on the first circle from the radicalaxis bears a, constant ratio to the square of the length of the tangentdrawn from that point to the second circle. 5. Find the equation of a system of circles which have the line x-y = 0for their radical axis. 6. AH circles represented by the equation {x-a^r+{y-b,f-c^^=\l{x-a^)^+{y-b^f-c^^have a common
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