The curves ${{C}_{1}}$ and ${{C}_{2}}$ have equations $y=\frac{2{{x}^{2}}+9}{{{x}^{2}}-4}$ and

Question

The curves ${{C}_{1}}$ and ${{C}_{2}}$ have equations $y=\frac{2{{x}^{2}}+9}{{{x}^{2}}-4}$ and $\frac{{{x}^{2}}}{9}+\frac{{{y}^{2}}}{25}=1$ respectively. Find the equations of the asymptotes of the curve ${{C}_{1}}$.[3] Sketch ${{C}_{1}}$ and ${{C}_{2}}$ on the same diagram, stating the equations of any asymptotes, coordinates of any points where ${{C}_{1}}$ or ${{C}_{2}}$ crosses the axes and any turning points.[5] Find the $x$-coordinates of the points were the two curves intersect.[2] Hence solve the inequality $-5\sqrt{1-\frac{1}{9}{{x}^{2}}}\le \frac{2{{x}^{2}}+9}{{{x}^{2}}-4}$.[2]

Accepted Answer

(i) $x=-2$, $x=2$, $y=2$ (iii) $-1.19$ and $1.19$ (iv) $-1.19\le x\le 1.19$ OR $-3\le x < -2$ OR $2 < x\le 3$

2021 CJC Promo Q10

2021 CJC Promo Q10

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