Count ellipse intersections without losing the domain · source reference

Printed question

The curves $C_1,C_2$ have equations $(x-2)^2=a^2(1-y^2)$ and $y=\frac{x^2-4}{x+1}$, where $1<a<2$, respectively. Describe the geometrical shape of $C_1$. [1]

(a) State a sequence of transformations which transforms the graph of $x^2+y^2=1$ to the graph of $C_1$. [3]

(b) (i) Sketch $C_1,C_2$ on the same diagram, stating the coordinates of any points of intersection with the axes and the equations of any asymptotes. [6]

(ii) Show algebraically that the $x$-coordinates of the points of intersection of $C_1,C_2$ satisfy the equation $(x+1)^2(x-2)^2=a^2(x+1)^2-a^2(x^2-4)^2$. [2]

(iii) Deduce the number of real roots of the equation in part (ii).

[2010/HC/Prelim/I/12]

Printed working

$(x-2)^2=a^2(1-y^2)\Rightarrow\frac{(x-2)^2}{a^2}+y^2=1$.

$C_1$ is an ellipse with centre $(2,0)$, whose length of semi-major axis along the $x$-direction is $a$ units and length of semi-minor axis along the $y$-direction is 1 unit.

(a) Method 1 (Scaling – Translation):

1. Scale parallel to $x$-axis by factor $a$.
2. Translate in the positive $x$-direction by 2 units.

Method 2 (Translation – Scaling):

1. Translate in the positive $x$-direction by $\frac2a$ units.
2. Scale parallel to $x$-axis by factor $a$.

(b)(i) Printed graph below. Note: $1<a<2$.

(ii) At the points of intersection of $C_1,C_2$:

Solving $(x-2)^2=a^2(1-y^2)$ and $y=\frac{x^2-4}{x+1}$ simultaneously,

$(x-2)^2=a^2[1-(\frac{x^2-4}{x+1})^2]$.

$\Rightarrow(x-2)^2(x+1)^2=a^2[(x+1)^2-(x^2-4)^2]$.

$\Rightarrow(x+1)^2(x-2)^2=a^2(x+1)^2-a^2(x^2-4)^2$ (Shown).

(iii) 2 intersection points $\Rightarrow2$ real roots.

Printed answer key

$C_1$ is an ellipse with centre (2,0), semi-major axis a units along x and semi-minor axis 1 along y. (i) Both transformation methods transcribed above. (ii) Native graph above. (iii) 2 real roots. Short-key lettering (i),(ii),(iii) corresponds to the question’s (a),(b)(i),(b)(iii).

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.