State a sequence of transformations that will transform the curve with equation \({{y}^{2}}-{{x}^{2}

Question

State a sequence of transformations that will transform the curve with equation ${{y}^{2}}-{{x}^{2}}=1$ on to the curve with equation $9{{y}^{2}}-54y-{{x}^{2}}-2x+79=0$.[4] A curve $C$ has equation $9{{y}^{2}}-54y-{{x}^{2}}-2x+79=0$. For real values $x$, use a non-graphical method to determine that $y$ cannot lie between $a$ and $b$, where $a$ and $b$ are exact real constants to be determined.[3] Sketch the curve $C$, indicating clearly the equations of all asymptotes and the coordinates of the turning points.[3] By adding a suitable curve, determine the number of real roots of the equation, $9{{\left[ {{\left( x+1 \right)}^{2}}+3 \right]}^{2}}-54\left[ {{\left( x+1 \right)}^{2}}+3 \right]-{{x}^{2}}-2x+79=0$.[2]

Accepted Answer

(b)(i) $a=\frac{8}{3}$ and $b=\frac{10}{3}$. Thus $y$ cannot lie between $\frac{8}{3}$ and $\frac{10}{3}$. (iii)$2$

2024 ASRJC P1 Q7

2024 ASRJC P1 Q7

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