A square is inscribed in the circle x²+y²−10x−6y+30=0 with one side parallel to y=x+3. Find Σ(xᵢ²+yᵢ²) over the four vertices, using the fact that the square’s diagonals lie along the horizontal and vertical through the centre.
A line segment AB of length λ moves with both endpoints on a circle of radius λ. Find the radius of the locus of the point dividing AB in ratio 2:3, using the fixed subtended angle and the section formula.
Circle C1: x²+y²−2(x+y)+1=0 and C2 have centre (−1,0), radius 2. Their common chord meets the y-axis at P. Find the square of the distance from P to the centre of C1, using the radical axis.
A circle with centre (2,3) and radius 4 intersects x+y=3 at P and Q. Tangents at P, Q meet at S(α,β). Find 4α−7β using the chord of contact and the fact that it’s the same as the given line.
Given the line ax+y=c is tangent to both the circle x²+y²=1 and the parabola y²=4√2x, find |c| by combining the circle’s tangent-distance condition with the parabola’s tangent-in-slope-form condition.
Find the equation of a common tangent to the circle x²+y²−6x=0 and the parabola y²=4x, by writing the parabola’s tangent in slope form and applying the circle’s tangent-distance condition.
A circle with radius r<8 has 3x+4y=24 and 3x−4y=32 as tangents and 4x+3y=1 as a normal. Find α−β+r using the tangent-distance formula and the fact that a normal passes through the centre.