The maths education in Shanghai has a reputation for its high achievement and severe pressure. Well, I won’t say all the questions in their Y12 exam papers are unapproachable. Here is one I picked from the 2007 University Entrance Examination (the equivalent of HSC in that context), maths paper, and I found it seemed quite similar to the following HSC Mathematics question.
And the question in the Shanghai paper is like this:
If a curve is composed of half the ellipse x²/a²+y²/b²=1 (x≥0) and half the ellipse y²/b²+x²/c²=1 (x≤0), we may call it a “fruit circle”, where a²=b²+c², a>0, b>c>0. Given F_0, F_1, F_2 are the focus of the corresponding ellipses, while A_1, A_2 and B_1, B_2 are the intercepts of the “fruit circle”,
Q1. If the three focus, F_0, F_1 and F_2 are the vertices of an equilateral triangle with sides of length 1, find the equation for the “fruit circle”;
Q2. If |A_1A_2|>|B_1B_2|, find the range of the ratio b/a;
Q3. If a line intersects the “fruit circle” in two distinct points, the interval jointing these two points shall be named a chord of the fruit circle. By drawing a set of paralell chords, we can generate their midpoints accordingly. Is it possible that all the midpoints are on the locus of an ellipse? If so, please find all the possible gradients of the chords; if not, please give your reasons.
This is not an easy question. Actually, this is the last question of the paper, and the paper is kind of the equivalent of 4 Units Maths Paper. But even so, I think we still can solve the first two questions.
However, the last question is a real challenge. The answer is “if and only if the gradient is zero, will all the midpoints being positioned on an ellipse.” The answer seems trivial, but the discussion is not. See if you can solve it yourself. And don’t worry if you can’t. Even if a student couldn’t solve all the questions, they may still be enrolled by top universities.
Until next time.
Wen
Maths for us 