A curve $C$ has equation $4{{\left( x-k \right)}^{2}}=k\left( {{y}^{2}}-4 \right)$, where $k$ is a p

Question

A curve $C$ has equation $4{{\left( x-k \right)}^{2}}=k\left( {{y}^{2}}-4 \right)$, where $k$ is a positive constant. Sketch $C$, indicating clearly the equations of asymptotes, coordinates of turning points and axial intercepts. Describe a sequence of transformations that maps the graph of ${{y}^{2}}-{{x}^{2}}=4$ onto $C$.

Accepted Answer

Asymptotes: $y=\pm\frac{2}{\sqrt{k}}(x-k)$; turning points: $(k,\pm2)$; $y$-intercepts: $\left(0,\pm2\sqrt{k+1}\right)$; no $x$-intercepts. Transform by a scale factor $\frac{\sqrt{k}}2$ parallel to the $x$-axis, followed by a translation of $k$ units in the positive $x$-direction.

2014 HCI Promo Q5

2014 HCI Promo Q5

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