June 2025 Paper 3 Q12

12

The questions in this section refer to the article on the Insert. You should read the article before attempting the questions.

The relevant parts of the article “The trisectrix of Maclaurin” are reproduced below; the line numbers are those printed on the Insert.

Lines 6–7
In 1742, Colin Maclaurin first studied a curve which can be used to trisect an angle. The curve is called the Trisectrix of Maclaurin.

Lines 8–9
The equation of the curve in cartesian form is \(y^2 = \dfrac{x^2(3a-x)}{a+x}\), where \(a\) is a constant. The curve is shown in Fig. C2 below. Only values of \(a > 0\) are considered in this article.

Lines 10–11
It can be shown that everywhere on the curve \(\dfrac{3a-x}{a+x} \geqslant 0\). It follows that \(3a - x \geqslant 0\) and \(a + x > 0\).

Line 12
The curve has the trisection property illustrated in Fig. C2.

Lines 14–16
The point C has coordinates \((2a, 0)\) and Q is the point where the curve crosses the positive \(x\)-axis. The point P is a general point \((x, y)\) on the loop of the curve. The origin of the coordinate system is denoted by O.

Line 17
The dashed line is an asymptote to the curve.

Lines 18–19
If the line CP makes an angle \(3\theta\) with the positive \(x\)-axis then the line OP makes an angle \(\theta\) with the positive \(x\)-axis. Angles are measured anticlockwise from the positive \(x\)-axis.