tmua.me Tau Paper 1 · Question 13

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tmua.me Tau Paper 1 · Q13

Paper 1GeometryAlgebra

For a positive real number $c$ , let $C$ be the curve

x+y+xy=c(x^2+y^2).

For every $t$ with $0\le t\le1$ , the line $x=t$ meets $C$ at a point with $0\le y\le1$ .

What is the greatest possible value of $(x-y)^2$ at a point of $C$ satisfying $0\le x\le1$ and $0\le y\le1$ ?

\dfrac45

\dfrac{3-\sqrt5}{2}

2-\dfrac{4\sqrt5}{5}

\dfrac12

4-\dfrac{8\sqrt5}{5}

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