A curve $C$ has equation $y = \frac{x^2 + 25x + 114}{x + 1}$. Sketch $C$, indicating on the di

Question

A curve $C$ has equation $y = \frac{x^2 + 25x + 114}{x + 1}$. Sketch $C$, indicating on the diagram the equations of any asymptotes, and the coordinates of any turning points and any axial intercepts.[4] A curve $D$ has parametric equations $$x = 23\sin 2\theta - 1,\ y = 23 + 23\cos 2\theta\text{ for}\hspace{0.5em} -\frac{\pi}{2} < \theta < \frac{\pi}{2}.$$ Find the cartesian equation of curve $D$.[2] Hence, on the same diagram as in part (a), sketch the curve $D$ and determine the number of intersection points between curve $C$ and curve $D$.[3]

Accepted Answer

(a) Asymptotes $x=-1$ and $y=x+24$; turning maximum $(-1-3\sqrt{10},23-6\sqrt{10})$ and minimum $(-1+3\sqrt{10},23+6\sqrt{10})$; intercepts $(-19,0)$, $(-6,0)$, and $(0,114)$ (b) $(x+1)^2+(y-23)^2=23^2$, excluding $(-1,0)$ (c) $4$ intersection points

2024 EJC P1 Q4

2024 EJC P1 Q4

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