Spicy Circles Questions - Coordinate Geometry - A-level Mathematics

Questions answered in the video: 1. A circle with centre P and radius r touches externally both the circles x^2+y^2=4 and x^2+y^2-6x+8=0. Prove that the x-coordinate of P is 1/3r+2, and that P lies on the curve y^2=8(x-1)(x-2) 2. The circle S1 with centre C1(a1,b1) and radius r1 touches externally the circle S2 with centre C2(a2,b2) and radius r2. The tangent at their common point passes through the origin. Show that (a1^2-a^2)^2 + (b1^2-b2^2)=(r1^2-r2^2) If, also, the other two tangents from the origin to S1 and S2 are perpendicular, prove that |a2b1-a1b2|=|a1a2+b1b2| Hence show that, if C1 remains fixed but S1 and S2 vary, then C2 lies on the curve (a1^2-b1^2)(x^2-y^2)+4a1b1xy=0 3. Show that the points (-1,0) and (1,0) are on the same side of the line y=x-3. Find the equations of the two circles each passing through the points (-1,0), (1,0) and touching the line y=x-3. 4. Find the equation of the circle S which passes through A(0,4) and B(8,0) and has its centre on the x-axis. If the point C lies on the circumference of S, find the greatest possible area of triangle ABC. ❤️ ❤️ ❤️ Support the channel ❤️ ❤️ ❤️    / @mathonify