Figure 3 shows a sketch of the circle C with centre N and equation $$(x - 2)^2 + (y + 1)^2 = \frac{169}{4}$$ (a) Write down the coordinates of N - Edexcel - A-Level Maths Pure - Question 9 - 2010 - Paper 4

The radius can be found from the equation of the circle:

$r = \sqrt{\frac{169}{4}} = \frac{13}{2} = 6.5.$

Since chord AB is parallel to the x-axis and lies below the x-axis with a length of 12 units, we can set the coordinates as follows:

1. Let the y-coordinate of points A and B be $y_A = y_B = -4$.

2. To find the x-coordinates:

   • For point A, $x_A = 2 - 6 = -4$.
   • For point B, $x_B = 2 + 6 = 8$.

Thus, the coordinates are A(-4, -4) and B(8, -4).

Using the sine rule in triangle ANB, we have:

$\sin(\angle ANB) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{6}{6.5}.$

Calculating gives:

$\angle ANB = \arcsin\left(\frac{6}{6.5}\right) = 67.3°.$
Since we know the triangle's angle sum property, we can calculate:

$\angle ANB = 180° - 2 \times 67.3° = 134.8°.$

Using triangle ANP:

$AP = 6.5 \cdot \sin(67.4°)$ gives:

$AP \approx 15.6.$
Thus, the length of AP rounded to 3 significant figures is approximately 15.6.