Fit an elliptical tunnel from boundary points · source reference

Printed question

The diagram below shows the cross-section of a single track tunnel entrance for steam-engine in a model train system. The elliptical entrance of the tunnel can be modelled by the equation $x^2+Ay^2+By+C=0$.

The highest point of the tunnel entrance is 7 cm directly above the origin at ground level. The track, mounted on track bed $PS$, is centred about the origin. A train carrying containers which measured 2 cm wide may stack goods as high as 6.8 cm.

Solve for the constants $A,B,C$, to 4 decimal places. Hence find the length of the widest part of the tunnel, to the nearest 0.1 cm. [6]

[2008/ACJC/Prelim/II/2]

Printed working

$x^2+Ay^2+By+C=0$.

Sub $(2.15,0)$, $C=-2.15^2=-4.6225$.

Sub $(1,6.8)$, $1+6.8^2A+6.8B+C=0$.

$46.24A+6.8B=3.6225$.

Sub $(0,7)$, $49A+7B+C=0$.

$49A+7B=4.6225$.

Solving, $A=0.6381827731=0.6382$ (4 d.p.),

$B=-3.806922269=-3.8069$ (4 d.p.).

$x^2+0.63818y^2-3.8069y-4.6225=0$.

$x^2+0.63818(y^2-5.9653y)-4.6225=0$.

$x^2+0.63818[(y-2.9826)^2-8.8961]-4.6225=0$.

$x^2+0.63818(y-2.9826)^2=10.300$.

When $y=2.9826$, $x^2=10.300\therefore x=3.2093$.

Widest part of tunnel is 6.4 cm (to 1 d.p.).

Teacher notes

For train carrying containers with goods as high as 6.8m to pass through tunnel, the train must be centred at the origin (i.e. 1m on either side of origin along $Ox$ direction).

For ellipse, $\frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=1$, the widest part along $x$-direction is $|a|$ and it occurs when $(y-k)^2=0$ (i.e. along line of symmetry $y=k$).

Printed answer key

$A=0.6382,B=-3.8069,C=-4.6225$ (4 d.p.); 6.4 cm.

Editorial qualification

The teacher note prints “6.8m” and “1m”; the question and diagram use centimetres, so these are unit typos. Also $|a|$ is the semi-width of the ellipse; the full widest length is $2|a|$, which the printed final 6.4 cm uses correctly.

Source provenance

2025 TJC H2 Math (9758) JC2 MSM: EQUATIONS. Question sheet printed pages 15–19; solution sheet printed pages 1–12. Original exam attribution is retained above. Currency dollar signs are spelled out outside maths to avoid confusing currency with TeX delimiters.