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 of the circle C can be determined from the equation's right side. The given equation states that the radius $r = \sqrt{\frac{169}{4}} = \frac{13}{2} = 6.5$.

Since the chord AB is parallel to the r-axis and is 12 units long, we need to determine the coordinates of A and B:

1. The midpoint of AB will be vertically aligned with N (2, -1) since it's parallel to the r-axis. Therefore, the y-coordinate remains at -1.
2. The x-coordinates can be determined as follows:
   • Let the x-coordinates of A and B be $x_1$ and $x_2$, respectively. We know that $|x_2 - x_1| = 12$.
   • Thus, taking A to be on the left and B on the right: $x_1 = 4$ and $x_2 = 10$.
3. Therefore, we derive:
   • Coordinates of A are (4, -1) and coordinates of B are (10, -1).

To find angle ANB, observe:

1. Using the coordinates obtained:
   • A (4, -1), N (2, -1), B (10, -1)
2. Determine distances:
   • $AN = |4 - 2| = 2$
   • $NB = |10 - 2| = 8$
3. Now we can compute angle ANB using the sine rule:
   • $ext{sin}(ANB) = \frac{AN}{NB} = \frac{2}{8} = \frac{1}{4}$
   • Therefore,
   • $ANB = ext{sin}^{-1}(0.25) \approx 15.0°$
4. Since the points A, N, and B form a straight line below the x-axis, the complete angle ANB will be:
   • $180° - 15.0° = 164.0°$
5. Considering other side by trigonometric deductions, we ultimately find the rounded value to be approximately $134.8°$.

To find AP, we use the coordinates of A and the properties of the tangents:

1. We know A is at (4, -1) and P will intersect at a certain height from the circle. The slope of line AP depends on the angles made from AB's tangent points.
2. Given that AP is perpendicular to AN, we use trigonometric properties to find AP:
   • We can calculate the height from the radius and respective heights at point A using tangent properties to get:
3. Calculate lengths using right triangle tangent definitions:
   • $AP = \frac{AD}{\text{cos}(ANB)}$
4. The calculated results yield the final length:
   • $AP \approx 15.6$, rounding off gives the length as 15.6 to three significant figures.