This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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You're on a roll — Step 1: Show that M is the point (3; -1) (for 10.1) The equation of the circle is given as x^2 + y^2 - 6x + 2y - 8 = 0. To find the center M, we complete the square: (x^2 - 6x) + (y^2 + 2y) = 8 (x^2 - 6x + 9) + (y^2 + 2y + 1) = 8 + 9 + 1 (x - 3)^2 + (y + 1)^2 = 18 From this standard form (x-a)^2 + (y-b)^2 = r^2, the center M is (a; b) = (3; -1). Thus, M is the point (3; -1). Step 2: Determine the equation of MR (for 10.2) M is (3; -1) and R is (0; -4). First, calculate the gradient of MR: m_MR = (y_R - y_M)/(x_R - x_M) = (-4 - (-1))/(0 - 3) = (-3)/(-3) = 1 Now, use the point-slope form y - y_1 = m(x - x_1) with R(0; -4) and m=1: y - (-4) = 1(x - 0) y + 4 = x The equation of MR is y = x - 4. Step 3: Show that q = 2 - p (for 10.3) We are given that RMN = 90^. This means the line segment MR is perpendicular to the line segment MN. From Step 2, the gradient of MR is m_MR = 1. Since MR MN, the gradient of MN, m_MN, must be the negative reciprocal of m_MR: m_MN = -(1)/(m_MR) = -(1)/(1) = -1 Now, calculate m_MN using M(3; -1) and N(p; q): m_MN = (y_N - y_M)/(x_N - x_M) = (q - (-1))/(p - 3) = (q + 1)/(p - 3) Set this equal to -1: (q + 1)/(p - 3) = -1 q + 1 = -(p - 3) q + 1 = -p + 3 q = -p + 3 - 1 Thus, q = 2 - p. Step 4: Determine the values of p and q (for 10.4) N(p; q) lies on the circle (x - 3)^2 + (y + 1)^2 = 18. Substitute (p; q) into the equation: (p - 3)^2 + (q + 1)^2 = 18 (*) From Step 3, we have q = 2 - p. Substitute this into equation (*): (p - 3)^2 + ((2 - p) + 1)^2 = 18 (p - 3)^2 + (3 - p)^2 = 18 Since (3 - p)^2 = (p - 3)^2: 2(p - 3)^2 = 18 (p - 3)^2 = 9 Take the square root of both sides: p - 3 = ±sqrt(9) p - 3 = ± 3 Case 1: p - 3 = 3 p = 6. Substitute p=6 into q = 2 - p: q = 2 - 6 = -4. So N is (6; -4). Case 2: p - 3 = -3 p = 0. Substitute p=0 into q = 2 - p: q = 2 - 0 = 2. So N is (0; 2). From the diagram, N is in the fourth quadrant (positive x, negative y). Therefore, the correct coordinates for N are (6; -4). The values are p = 6 and q = -4. Step 5: Determine the equation of the circle having centre O and passing through N (for 10.5) The center O is (0; 0). The circle passes through N(6; -4) (from Step 4). The radius squared, r_O^2, is the distance from O to N squared: r_O^2 = (x_N - x_O)^2 + (y_N - y_O)^2 r_O^2 = (6 - 0)^2 + (-4 - 0)^2 r_O^2 = 6^2 + (-4)^2 r_O^2 = 36 + 16 = 52 The equation of the circle with center O(0; 0) is x^2 + y^2 = r_O^2: x^2 + y^2 = 52 Step 6: Calculate the area of the circle centred at M (for 10.6) From Step 1, the equation of the circle centred at M is (x - 3)^2 + (y + 1)^2 = 18. The radius squared for this circle is r^2 = 18. The area of a circle is given by the formula A = r^2. A = (18) The area of the circle centred at M is 18 units^2. Step 7: Calculate the ratio (NP)/(MP) in its simplest form (for 10.7) M is the center (3; -1). R is (0; -4). N is (6; -4). The tangents drawn to the circle at R and N meet at P. We know MR and MN are radii, so MR = MN = sqrt(18) (from Step 1, r^2=18). Also, RMN = 90^ (given). A tangent to a circle is perpendicular to the radius at the point of tangency. So, MR RP and MN NP. This means MRP = 90^ and MNP = 90^. Since all four angles of quadrilateral MRNP are 90^ ( RMN=90^, MRP=90^, MNP=90^, and thus RPN=90^), MRNP is a rectangle. Because MR = MN = sqrt(18), it is a square. In a square, all sides are equal, so NP = MR = sqrt(18). The diagonals of a square are equal. So MP = RN. Calculate the length of RN: RN = (x_N - x_R)^2 + (y_N - y_R)^