In the diagram, N is the centre of the circle - NSC Mathematics - Question 4 - 2017 - Paper 2

The center of the circle is at N(-1, 1) and the radius is the distance from N to M or P. Using the distance formula:

$r = \sqrt{((-3) - (-1))^2 + ((-2) - 1)^2} = \sqrt{(2)^2 + (-3)^2} = \sqrt{4 + 9} = \sqrt{13}$

Thus, the equation of the circle is:

$(x - (-1))^2 + (y - 1)^2 = (\sqrt{13})^2$

Simplifying gives:

$(x + 1)^2 + (y - 1)^2 = 13$.

The slope of MR can be calculated using the coordinates of R (which we will find later) and M(-3, -2). Let's denote the coordinates of R as (x_R, y_R).

The slope ( m_{RM} ) is given by:

$m_{RM} = \frac{y_R - (-2)}{x_R - (-3)} = \frac{y_R + 2}{x_R + 3}$

Using the property of tangents:

$m_{RM} \times m_{NM} = -1$

From our calculations of N to M, we find the slope of NM:

$m_{NM} = \frac{1 - (-2)}{-1 - (-3)} = \frac{3}{2}$

Thus:

$\frac{y_R + 2}{x_R + 3} \times \frac{3}{2} = -1$

This will allow us to express R's coordinates in terms of x_R.

Since the line joining S to M is perpendicular to the x-axis, this means S must have the same x-coordinate as M, which is -3. Thus, we can write:

$S = (-3, y_S)$

To find y_S: Since S lies on the circle, substituting into the circle equation:

$(x + 1)^2 + (y - 1)^2 = 13$

Substituting x = -3:

$( -3 + 1)^2 + (y_S - 1)^2 = 13 \Rightarrow ( -2)^2 + (y_S - 1)^2 = 13 \Rightarrow 4 + (y_S - 1)^2 = 13$

Solving gives:

$(y_S - 1)^2 = 9 \Rightarrow y_S - 1 = \pm 3 \Rightarrow y_S = 4 \text{ or } y_S = -2.$

Given that R is equidistant to the points where the tangents meet, the coordinates can be found by equating the lengths from R to the points where the tangents meet. Assuming y-coordinate from the earlier calculations:

$x_R^2 + (y_R - 4)^2 = (x_R + 3)^2 + (y_R + 2)^2$

This leads to an equation that can be solved for the coordinates of R.

The area of quadrilateral RSNM can be found using the formula for the area of a kite or by any suitable area formula for non-regular quadrilaterals. Given that RM is a diagonal and SM is another:

$\text{Area}_{RSNM} = \frac{1}{2} \times (MS \times RN)$

Substituting in the known lengths will provide the area.