Graphs and transformations: NYJC 2024 Promo Q3

Nanyang Junior College2024 PromoPromo Q3●●○ Standard6 marks

What this question tests

Graphing techniques.

Question

The curve $D$ has equation $y = \dfrac{2x - 5}{x^2 + 2x - 3}$ .

Sketch the graph of $D$ , stating the equations of any asymptotes, the coordinates of the points where the curve crosses the axes and the stationary point(s). [3]
By considering a suitable curve, hence, find the exact range of values of $k$ , where $k > 0$ , such that the equation $(x + 3)^2 + \left(\dfrac{2x - 5}{x^2 + 2x - 3}\right)^{\!2} = k^2$ has at least one positive real root. [3]

Show full worked solution▾

(a)

$Nanyang Junior College 2024 Promo Q3: solution diagram$

(b)

$\begin{aligned} (x+3)^2+\left(\frac{2x-5}{x^2+2x-3}\right)^2&=k^2\\ (x+3)^2+y^2&=k^2\\ k&=\sqrt{\left(\frac53-0\right)^2+\left(0-(-3)\right)^2}\\ &=\sqrt{\frac{25}{9}+9}\\ &=\sqrt{\frac{106}{9}}\\ &=\frac{\sqrt{106}}{3} \end{aligned}$ $\therefore k>\frac{\sqrt{106}}{3}$

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