A squared inequality is a region between two branches · source reference

Printed question

The curve $C_1$ has equation $\frac{y^2}{4}-x^2=1$ and the curve $C_2$ has equation $y=k-\frac1x$, where $k$ is a positive constant. The horizontal asymptote of $C_2$ is a tangent to $C_1$.

(i) Sketch $C_1$, giving the equations of any asymptotes and the coordinates of any turning points. [3]

(ii) Explain why $k=2$. [1]

(iii) On the same diagram as $C_1$, sketch $C_2$ showing the coordinates of the points of intersection of $C_1,C_2$. [3]

(iv) It is given that if $[f(x)]^2\le g(x)$, then $-\sqrt{g(x)}\le f(x)\le\sqrt{g(x)}$. Using this result and the graphs in part (iii), solve the inequality $(2-\frac1x)^2\le4(x^2+1)$. [2]

[2016/DHS/Promo/10]

Printed working

(i) $\frac{y^2}{4}-x^2=1$.

(ii) $C_2:y=k-\frac1x$.

Horizontal asymptote $y=k$.

$y=k$ is a line with gradient 0. Since it is tangent to $C_1$, it must be the tangent to the points at $x=0$, which has tangent $y=2$ at $(0,2)$ and tangent $y=-2$ at $(0,-2)$. Since $k$ is positive, the tangent line involved must be $y=2$, i.e. $k=2$.

(iii) Printed intersections $(-1.07,2.93)$ and $(0.246,-2.06)$.

(iv) $(2-\frac1x)^2\le4(x^2+1)$.

Using the given result with $f(x)=2-\frac1x$ and $g(x)=4(x^2+1)$,

$-2\sqrt{x^2+1}\le2-\frac1x\le2\sqrt{x^2+1}$.

From graph in (iii), $x\le-1.07$ or $x\ge0.246$.

Teacher notes

1. To find asymptotes: Let $\frac{y^2}{4}-x^2=0\Rightarrow(\frac y2)^2=x^2\Rightarrow y=\pm2x$.

2. To find vertices: Put $x^2=0\Rightarrow x=0$. Vertices: $(0,-2),(0,2)$.

$\frac{y^2}{4}-x^2=1$ (hyperbola) $\Rightarrow y=2\sqrt{x^2+1}$, $y>0$; $y=-2\sqrt{x^2+1}$, $y<0$.

Printed answer key

(i) Native hyperbola. (ii) Native combined diagram [the short key labels the combined diagram (ii) though it answers question (iii)]. (iv) x≤−1.07 or x≥0.246.

Source provenance

2025 TJC H2 Math (9758) JC2 MSM: GRAPHS AND CURVE SKETCHING. Question/answer booklet printed pages 1–14; solutions printed pages 1–28. Running metadata and repeated headings are transcribed in the topic metadata article.