tmua.me Tau Paper 2 · Question 14

For each real number $h$ , let $\mathcal L_h$ be the set of distinct straight lines which pass through $(h,-1)$ and are tangent to

y=x^4-2x^2.

A line tangent at two different points is counted only once. If $\mathcal L_h$ has exactly three elements, let $S(h)$ be the sum of the squares of their gradients.

For a real number $C$ , consider the statement

\text{there are exactly four real numbers $h$ for which $|\mathcal L_h|=3$ and $S(h)=C$.}

Which condition is necessary and sufficient for this statement?

\displaystyle \frac{64(2-\sqrt2)}{27}<C\le\frac{128}{27},\quad C\ne\frac{1024}{729}

\displaystyle \frac{64(2-\sqrt2)}{27}<C<\frac{128}{27}

\displaystyle \frac{64(2-\sqrt2)}{27}<C<\frac{128}{27},\quad C\ne\frac{1024}{729}

\displaystyle \frac{64(2-\sqrt2)}{27}<C<\frac{128}{27},\quad C\ne\frac{512}{729}

\displaystyle \frac{64(2-\sqrt2)}{27}\le C<\frac{128}{27},\quad C\ne\frac{1024}{729}

examday

tmua.me Tau Paper 2 · Q14