Circle C passes through (4,0) and touches x²+y²+4x−6y=12 externally at (1,−1). Find C’s radius using the tangent-point-lies-on-the-line-of-centres property.
Two circles in the first quadrant with radii r₁, r₂ touch both coordinate axes, each cutting a 2-unit intercept on x+y=2. Find r₁²+r₂²−r₁r₂.
A circle x²+y²−αx+βy+γ=0 touches the x-axis at (a,0) and cuts a y-axis intercept of length b, lying below the x-axis. Find the ordered pair (2a, b²).
Three circles of radii a, b, c touch each other externally with the x-axis as a common tangent. Derive the relation between a, b, c using centre coordinates and distance.
Circle C passes through A(2,−1) and B(3,4), AB not a diameter, and its centre lies on (x−5)²+(y−1)²=13/2. Find r² using the perpendicular-bisector property.