Problems

1. Explain why the equation \((y - x + 3)(y + x - 5) = 0\) represents a pair of straight lines with gradients \(1\) and \(-1\). Show further that the equation \[y^2 - x^2 + py + qx + r = 0\] represents a pair of straight lines with gradients \(1\) and \(-1\) if and only if \(p^2 - q^2 = 4r\).

2. Explain why the coordinates of any point which lies on both of the curves \(C_1\) and \(C_2\) also satisfy the equation \[y^2 + 2sy + s(s+1) - x + k(y - x^2) = 0\] for any real number \(k\). Given that \(s\) is such that \(C_1\) and \(C_2\) intersect at four distinct points, show that choosing \(k = 1\) gives an equation representing a pair of straight lines, with gradients \(1\) and \(-1\), on which all four points of intersection lie.
3. Show that if \(C_1\) and \(C_2\) intersect at four distinct points, then \(s < -\frac{3}{4}\).
4. Show that if \(s < -\frac{3}{4}\), then \(C_1\) and \(C_2\) intersect at four distinct points.

Consider a fixed square \(ABCD\) and a variable point \(P\) in the plane of the square. We write the perpendicular distance from \(P\) to \(AB\) as \(p\), from \(P\) to \(BC\) as \(q\), from \(P\) to \(CD\) as \(r\) and from \(P\) to \(DA\) as \(s\). (Remember that distance is never negative, so \(p,q,r,s\geqslant 0\).) If \(pr=qs\), show that the only possible positions of \(P\) lie on two straight lines and a circle and that every point on these two lines and a circle is indeed a possible position of \(P\).