Standard +0.8 This is a Further Maths question requiring conversion to polar coordinates using x=r cos θ and y=r sin θ, then algebraic manipulation with trigonometric identities (specifically the double angle formula for sin 2θ). Part (ii) requires recognizing that maximizing r² involves minimizing the denominator. While the steps are systematic, it demands facility with polar coordinates and trig identities beyond standard A-level, plus some insight to connect the algebraic form to the geometric constraint.
A curve has equation \(x^4 + y^4 = x^2 + y^2\), where \(x\) and \(y\) are not both zero.
\begin{enumerate}[label=(\roman*)]
\item Show that the equation of the curve in polar coordinates is \(r^2 = \frac{2}{2-\sin^2 2\theta}\). [4]
\item Deduce that no point on the curve \(x^4 + y^4 = x^2 + y^2\) is further than \(\sqrt{2}\) from the origin. [2]
A curve has equation $x^4 + y^4 = x^2 + y^2$, where $x$ and $y$ are not both zero.
\begin{enumerate}[label=(\roman*)]
\item Show that the equation of the curve in polar coordinates is $r^2 = \frac{2}{2-\sin^2 2\theta}$. [4]
\item Deduce that no point on the curve $x^4 + y^4 = x^2 + y^2$ is further than $\sqrt{2}$ from the origin. [2]
</end{enumerate}
\hfill \mbox{\textit{OCR Further Pure Core 2 Q9 [6]}}