tmua.me Tau Paper 2 · Question 8

A circle meets the parabola $y=x^2$ at four distinct points $A,B,C,D$ , in increasing order of their $x$ -coordinates $a<b<c<d$ . Suppose

a^2+b^2+c^2+d^2=6

and the diagonals $AC$ and $BD$ are perpendicular. Exactly one of the points

X=\left(0,\frac14\right),\qquad Y=\left(0,\frac94\right)

lies strictly inside the quadrilateral $ABCD$ . A point on its boundary does not count as strictly inside.

Which set is exactly the set of possible areas of $ABCD$ ?

\displaystyle [3,9-3\sqrt3]\ \cup\ [1+\sqrt{11},5]

\displaystyle (3,9-3\sqrt3]\ \cup\ [1+\sqrt{11},5]

\displaystyle [9-3\sqrt3,1+\sqrt{11}]

\displaystyle (3,5]\setminus\{9-3\sqrt3,1+\sqrt{11}\}

\displaystyle (3,9-3\sqrt3)\ \cup\ (1+\sqrt{11},5]

examday