The curve $C$ has equation ${{\left( x-2 \right)}^{2}}-{{\left( y-3 \right)}^{2}}=1$. Sketch \(C

Question

The curve $C$ has equation ${{\left( x-2 \right)}^{2}}-{{\left( y-3 \right)}^{2}}=1$. Sketch $C$, stating the equations of the asymptotes and their point of intersection.[3] By considering the graph in part (i) and the straight line $y=mx-2m+3,$ find the range of values of $m$ such that the equation ${{\left( x-2 \right)}^{2}}-{{\left( mx-2m \right)}^{2}}=1$ has no real roots. Explain why it is impossible for the equation to have only one real root.[3]

Accepted Answer

(i) Asymptotes $y=x+1$ and $y=-x+5$, centre $(2,3)$, vertices $(1,3)$ and $(3,3)$ (ii) $m\le-1$ or $m\ge1$, and one real root is impossible

2015 JJC P1 Q2

2015 JJC P1 Q2

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