. Algebraic geometry; a new treatise on analytical conic sections . at the point {h, h). BXg. VII. a.] ORTHOGONAL CIRCLES. 105 7. Three circles are described of equal radii (r), having their centresat the points (0, 0), (0, b), (a, 0). Find the equation of the circle whichcuts them all orthogonally. 8. Find the general equation of the circles which cut the circle x^ + y^-2x + 2y-2=0at right angles at the point (1, 1). 9. Find the condition that the circles x + y^-2ax-c=0, ofi+y^+2bx-c=0may cut orthogonally. 10. Find the equation of the circle which passes through the origin andcuts orthogonall
. Algebraic geometry; a new treatise on analytical conic sections . at the point {h, h). BXg. VII. a.] ORTHOGONAL CIRCLES. 105 7. Three circles are described of equal radii (r), having their centresat the points (0, 0), (0, b), (a, 0). Find the equation of the circle whichcuts them all orthogonally. 8. Find the general equation of the circles which cut the circle x^ + y^-2x + 2y-2=0at right angles at the point (1, 1). 9. Find the condition that the circles x + y^-2ax-c=0, ofi+y^+2bx-c=0may cut orthogonally. 10. Find the equation of the circle which passes through the origin andcuts orthogonally the circles whose equations are x+y-6x + S=0, x^ + y-2x-2y=1. 11. Find the locus of the centres of circles which cut the circles x^+y^+4:X-6y + Q=0, x + y^-4:X + 6y+4:=Qorthogonally. 12. A circle, whose centre is in the axis of x, cuts the circles sii + y^-6y + 5=0, x^+y^-i-6x-31=Qorthogonally. Find its equation. RADICAL AXIS OF TWO CIRCLES. 107. Def. The locus of a point, which moves so that thetangents drawn from it to two given fixed circles are equal, is a. Pig. 68. straight Jine perpendicular to the line of centres of the circles,and is called their Eadical Axis. Let {x-af + {y-fiy = a\ {x-o!f + {y-Pf = a^ be the equa-tions of the circles. 106 RADlCAti AXIS OF TWO CIECLES. [chap. vii. Then if P{x^, y^) is any point on the locus, («! - a)2 + (y^ - /3)2 - a2 = (a;i - a)^ + {y^ ^ ^J - a2 (Art. 99) or 1x^ (a - a) + 1y^ (ff - ^) + a^ + ^^ - a2 - /82 - a^ + a2 = 0. .. suppressing suflSxes, 2a; (a - a) + 2y (;S _ /3) + a2 + ^2 - a2 - /32 - a^ _ a2 = Q is the equation of the locus. Being of the first degree, this is a straight line. Its slope = - -ai 75- The slope of J;he line of centres = —,——. a — a The product of these slopes= - 1; {mm! = - 1) .. the radical axis is per-pendicular to the line ofcentres. Note 1. When the coefficientsof x^ and y are unity in the equa-tions of two circles, the equationof their radical axis is at onceobtained by subtraction. Note 2. When
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