A curve $D$ has equation \(\frac{{{\left( x+2 \right)}^{2}}}{{{2}^{2}}}-{{\left( y-1 \right)}^{2}}

Question

A curve $D$ has equation $\frac{{{\left( x+2 \right)}^{2}}}{{{2}^{2}}}-{{\left( y-1 \right)}^{2}}=1$. Another curve $E$ has equation ${{x}^{2}}-{{y}^{2}}=1$. Describe a sequence of transformations that transforms the graph of $E$ onto $D$.[3] Find the equations of the asymptotes of $D$.[2] Sketch $D$, stating the equations of the asymptotes and the coordinates of the vertices.[3] Given an ellipse with equation ${{\left( \frac{x+2}{c} \right)}^{2}}+{{\left( y-1 \right)}^{2}}=1$, state the possible values of $c$ such that the ellipse does not intersect $D$.[1]

Accepted Answer

(i) First stretch parallel to the $x$-axis by scale factor $2$, then translate by $\begin{pmatrix}-2\1\end{pmatrix}$ (ii) $y=\dfrac{x}{2}+2$ and $y=-\dfrac{x}{2}$ (iii) Vertices $(-4,1)$ and $(0,1)$, opening left and right with the stated asymptotes (iv) $-2<c<0$ or $0<c<2$

2023 NJC Promo Q2

2023 NJC Promo Q2

Video Solution:

Solution:

Solution locked