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Graphs and transformations: ASRJC 2024 Promo Q1

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Anderson Serangoon Junior College2024 PromoPromo Q1●●○ Standard4 marks

What this question tests

Conics & Equations.

Question

A hyperbola has the equation $a(x-2)^2 - b(y+1)^2 = c$ , where $a$ , $b$ and $c$ are positive integers. It passes through the point $(7, 10)$ . A line $y = x - 1$ intersects the hyperbola at $x = 1$ . Find one possible value for each of $a$ , $b$ and $c$ . [4]

Show full worked solution▾

Sub $(7,10) \Rightarrow 25a-121b-c=0$

Line $y=x-1$ cuts the graph at $x=1$

Sub $x=1$ and $y=0$ into equation of hyperbola $\Rightarrow a-b-c=0$

Using GC: $a=\frac54c,\quad b=\frac14c$ Sub into equation of hyperbola $\frac54c(x-2)^2-\frac14c(y+1)^2=c$ Since $a,b,c$ are positive integers $5(x-2)^2-1(y+1)^2=4\text{ is a possible equation of the hyperbola}$ Where $a=5,\quad b=1,\quad c=4$

Answers

$a = 5$ , $b = 1$ , $c = 4$ (one possible set)

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