91Ó°ÊÓ

Find the equation of the circle \(S\) which passes through \(A(0,4)\) and \(\mathrm{B}(8,0)\) and has its centre on the \(x\)-axis. If the point \(C\) lies on the circumference of \(S\), find the greatest possible area of the triangle \(A B C\). (U of L)

A circle with centre \(\mathrm{P}\) and radius \(r\) touches externally both the circles \(x^{2}+y^{2}=4\) and \(x^{2}+y^{2}-6 x+8=0 .\) Prove that the \(x\)-coordinate of \(\mathrm{P}\) is \(\frac{1}{3} r+2\), and that \(\mathrm{P}\) lies on the curve \(y^{2}=8(x-1)(x-2) .\) (U of \(\left.\mathrm{L}\right)\)

Find the equation of the two circles which each satisfy the following conditions: (a) the axis of \(x\) is a tangent to the circle, (b) the centre of the circle lies on the line \(2 y=x\), (c) the point \((14,2)\) lies on the circle. Prove that the line \(3 y=4 x\) is a common tangent to these circles.

If the normal at \(\mathrm{P}\left(a p^{2}, 2 a p\right)\) to the parabola \(y^{2}=4 a x\) meets the curve again at \(\mathrm{Q}\left(a q^{2}, 2 a q\right)\) prove that \(p^{2}+p q+2=0 .\) Prove that the equation of the locus of the point of intersection of the tangents to the parabola at \(\mathrm{P}\) and \(Q\) is $$ y^{2}(x+2 a)+4 a^{3}=0 $$

Show that the tangent at the point \(\mathrm{P}\), with parameter \(t\), on the curve \(x=3 t^{2}, \quad y=2 t^{3} \quad\) has equation \(y=t x-t^{3}\), Prove that this tangent will cut the curve again at the point \(Q\) with coordinates \(\left(3 t^{2} / 4,-t^{3} / 4\right)\). Find the coordinates of the possible positions of \(P\) if the tangent to the curve at \(P\) is the normal to the curve at \(Q\). \((\mathrm{U}\) of \(\mathrm{L})\)

If the line \(y=m x+2\) is a tangent to the parabola \(y^{2}=4 x\) there are two possible values of \(m\).