Challenging +1.2 This is a Further Maths question requiring implicit differentiation to find dy/dx, setting it to zero for stationary points, then finding d²y/dx² implicitly to determine nature. While it involves multiple steps and implicit second derivatives (which many students find challenging), the algebraic manipulation is straightforward once the method is known, and the question provides significant guidance by giving the relationship x = -2y upfront. Slightly above average difficulty due to the Further Maths context and second derivative requirement, but remains a standard textbook-style question.
8 A curve \(C\) has equation \(x ^ { 2 } + 4 x y - y ^ { 2 } + 20 = 0\). Show that, at stationary points on \(C , x = - 2 y\).
Find the coordinates of the stationary points on \(C\), and determine their nature by considering the value of \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) at the stationary points.
8 A curve $C$ has equation $x ^ { 2 } + 4 x y - y ^ { 2 } + 20 = 0$. Show that, at stationary points on $C , x = - 2 y$.
Find the coordinates of the stationary points on $C$, and determine their nature by considering the value of $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }$ at the stationary points.
\hfill \mbox{\textit{CAIE FP1 2016 Q8 [11]}}