Free · Year 3 of high school · 50 minutes · 15 questions · no penalty

Practice maths class test — Year 3: ellipse, hyperbola and homographic function.

15 questions drawn at random from a bank of 25, on the block of the programme your teacher is finishing: every attempt is a different test. No penalty for wrong answers, so answer even if you are not sure. At the end you receive by email the correct answer to every question and your weak topics — with an account you see them straight away here on the page. The step-by-step solution unlocks with your first lesson.

A class test is the written test an Italian high-school class sits after each block of the programme; this practice version is multiple choice.

Test in ItalianThe questions of this simulator are in Italian, as in the real exam.

Test prepared by Gaetano Livornese · Aerospace engineer, Politecnico di Milano · maths and physics tutor · updated 13 October 2026

Programme covered by the test

Standard equation of the ellipse, semi-axes and foci
Eccentricity and finding an ellipse from conditions
Line and ellipse: secant, tangent, external; the tangent-splitting formula
Standard equation of the hyperbola and asymptotes
Hyperbola referred to the axes and referred to the asymptotes
Homographic function: asymptotes, centre of symmetry, degeneration
Recognising the conic from its equation

Two worked examples from the test

Question 1 · Ellisse, iperbole, funzione omografica · Ellisse e iperbole

Determina le coordinate dei fuochi dell'ellisse di equazione

$\frac{x ^{2}}{169} + \frac{y ^{2}}{144} = 1$

A $F_{1} (0, - 5)$ , $F_{2} (0, 5)$
B $F_{1} (- 5, 0)$ , $F_{2} (5, 0)$ correct answer
C $F_{1} (- 313, 0)$ , $F_{2} (313, 0)$
D $F_{1} (- 13, 0)$ , $F_{2} (13, 0)$

Solution

L'equazione è già in forma canonica: $a^{2} = 169$ e $b^{2} = 144$ , quindi $a = 13$ e $b = 12$ .
Il semiasse maggiore è $a = 13$ e sta sull'asse $x$ : i fuochi si trovano su quell'asse.
Per l'ellisse vale $c^{2} = a^{2} - b^{2}$ :

$c^{2} = 169 - 144 = 25 \Rightarrow c = 5$

I fuochi sono quindi $F_{1} (- 5, 0)$ e $F_{2} (5, 0)$ .

Attenzione: la formula con il più, $c^{2} = a^{2} + b^{2}$ (che darebbe $313$ ), è quella dell'iperbole. Nell'ellisse i fuochi stanno dentro la curva, perciò deve venire $c < a$ .

Question 2 · Ellisse, iperbole, funzione omografica · Ellisse e iperbole

Scrivi in forma canonica l'ellisse di equazione

$16 x^{2} + 25 y^{2} = 400$

e calcolane l'eccentricità.

A $e = \frac{41}{5}$
B $e = \frac{4}{5}$
C $e = \frac{3}{5}$ correct answer
D $e = \frac{5}{3}$

Solution

Dividi tutto per il termine noto $400$ :

$\frac{16 x ^{2}}{400} + \frac{25 y ^{2}}{400} = 1 \Rightarrow \frac{x ^{2}}{25} + \frac{y ^{2}}{16} = 1$

Quindi $a = 5$ e $b = 4$ , con i fuochi sull'asse $x$ .
Semidistanza focale: $c^{2} = a^{2} - b^{2} = 25 - 16 = 9$ , cioè $c = 3$ .
Eccentricità: $e = \frac{c}{a} = \frac{3}{5} = 0, 6 .$

Per un'ellisse si ha sempre $0 \leq e < 1$ : un risultato maggiore o uguale a $1$ segnala che hai usato $c^{2} = a^{2} + b^{2}$ , cioè la formula dell'iperbole.

In the online practice test you will find 13 more questions like these; the correct answer to every question arrives by email.

Prefer paper? Download the test as a PDF (15 questions, 5 pages · 86 KB, ready to print). You receive the solutions by email after taking it online.

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What a maths test on ellipse and hyperbola looks like

The standard equation from given conditions (foci, semi-axes, eccentricity, passing through a point), a line and an ellipse in relative position, also with the tangent-splitting formula, the asymptotes of the hyperbola and a homographic function to study.

The mistake that costs the grade

In the ellipse, applying c² = a² − b² without looking at which semi-axis is larger: if the foci are on the y axis the relation is written the other way round. In the hyperbola, using that same formula instead of c² = a² + b².

How to use this practice test

Take it once cold, without revising and with the timer on: it is there to show you where you stand, not to look good. Then look at your weak topics, revise only those — usually two or three, not the whole chapter — and retake it after a few days. The 15 questions are drawn from a bank of 25, so the second attempt is not the same test and the scores are genuinely comparable.

Is the score not enough?

See which topics lost you the most marks, then read how to study maths without stress — or we look at your results together: the diagnostic is free.

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