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

The center of the circle is N(-1, 1) and we need to find the radius. The radius can be determined by the distance from N to either point M or P. Let's calculate the distance from N to M:

Using the distance formula: $r = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} = \sqrt{(-3 - (-1))^2 + (-2 - 1)^2} = \sqrt{(-2)^2 + (-3)^2} = \sqrt{4 + 9} = \sqrt{13}$

Thus, the equation of the circle can be written as: $(x + 1)^2 + (y - 1)^2 = 13.$

The point M lies on the circle N, and the tangent RM can be found using the point-slope form. The slope of line RM can be computed by taking the differences in y-coordinates and x-coordinates from points R and M.

Let the coordinates of R be (x_R, y_R). The slope $m$ of RM is given as: $m_{RM} = \frac{y_R - (-2)}{x_R - (-3)} = \frac{y_R + 2}{x_R + 3}$

Using the property of tangents to a circle where the tangent is perpendicular to the radius at the point of tangency (M), we find: $m_{RM} \cdot m_{NM} = -1$

Thus, compute the slope of NM and find R's coordinates accordingly.

Since the line joining S to M is perpendicular to the x-axis, this means that S must have the same x-coordinate as M. The coordinates of M are (-3, -2) therefore, S will have coordinates of the form (-3, y_S).

To find y_S, we will use the equation of the circle or the property of tangents. Given that the radius NM is vertical, the y-coordinate for S must provide a location where the tangent is drawn vertically. Thus, we use the calculated radius to find this exact point.

To find the coordinates of R, we equate the lengths of the tangents from R to points M and S. The distances from R(x_R, y_R) to M(-3, -2) and from R to S(-3, y_S) must equal:

$(x_R + 3)^2 + (y_R + 2)^2 = r^2$

Given that these tangents are equal by nature, we can solve the equations simultaneously to obtain the coordinates of R.

To calculate the area of the quadrilateral RSNM, we can split it into two triangles, RSN and RMN. The formula for the area of triangles can be used:

$Area = \frac{1}{2} \times base \times height$

Calculate area for both triangles separately, taking the bases as the distances between the relevant points and the height as determined earlier. Sum the areas to find the total area of RSNM.