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1999 Paper 2 Q7
D: 1600.0
B: 1500.0
curve sketching
stationary points
differentiation
quotient rule
discriminant analysis
asymptotes
rational function
parameter variation
The curve \(C\) has equation $$ y = \frac x {\sqrt{x^2-2x+a}}\; , $$ where the square root is positive. Show that, if \(a>1\), then \(C\) has exactly one stationary point. Sketch \(C\) when (i) \(a=2\) and (ii) \(a=1\).
Solution: \begin{align*} && y &= \frac x {\sqrt{x^2-2x+a}} \\ && y' &= \frac{\sqrt{x^2-2x+a} - \frac{x(x-1)}{\sqrt{x^2-2x+a}}}{x^2-2x+a} \\ &&&= \frac{-x+a}{(x^2-2x+a)^{3/2}} \end{align*} Since the denominator is always positive, the only stationary point is when \(x = a\)
