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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Alright Nqøbizithå🤍🩺 — let's do this. Here are the solutions for question 3: a) To determine the coordinates of A and B, we substitute the equation of the straight line y = x + 2 into the equation of the circle x^2 + y^2 = 20. Step 1: Substitute y = x + 2 into the circle equation. x^2 + (x + 2)^2 = 20 x^2 + (x^2 + 4x + 4) = 20 2x^2 + 4x + 4 - 20 = 0 2x^2 + 4x - 16 = 0 Divide by 2: x^2 + 2x - 8 = 0 Step 2: Factor the quadratic equation to find the x-coordinates. (x + 4)(x - 2) = 0 So, x = -4 or x = 2. Step 3: Substitute the x-values back into y = x + 2 to find the corresponding y-coordinates. For x = -4: y = -4 + 2 = -2. So, point A is (-4, -2). For x = 2: y = 2 + 2 = 4. So, point B is (2, 4). The coordinates of A and B are A(-4, -2) and B(2, 4). b) To determine the length of the chord AB, we use the distance formula between points A and B. Step 1: Apply the distance formula d = sqrt((x_2 - x_1)^2 + (y_2 - y_1)^2) for A(-4, -2) and B(2, 4). AB = sqrt((2 - (-4))^2 + (4 - (-2))^2) AB = sqrt((6)^2 + (6)^2) AB = sqrt(36 + 36) AB = sqrt(72) Step 2: Simplify the square root. AB = sqrt(36 × 2) AB = 6sqrt(2) The length of the chord AB is 6sqrt(2) units. c) To determine the coordinates of M, the midpoint of the chord AB, we use the midpoint formula. Step 1: Apply the midpoint formula M = ((x_1 + x_2)/(2), (y_1 + y_2)/(2)) for A(-4, -2) and B(2, 4). M = ((-4 + 2)/(2), (-2 + 4)/(2)) M = ((-2)/(2), (2)/(2)) M = (-1, 1) The coordinates of M are (-1, 1). d) To show that OM AB, we need to check if the product of their slopes is -1. O is the origin (0, 0). Step 1: Calculate the slope of OM. m_OM = (y_M - y_O)/(x_M - x_O) = (1 - 0)/(-1 - 0) = (1)/(-1) = -1 Step 2: Calculate the slope of AB. The line AB is y = x + 2, so its slope is 1. m_AB = 1 Step 3: Multiply the slopes. m_OM × m_AB = (-1) × (1) = -1 Since the product of the slopes is -1, OM is perpendicular to AB. OM AB. e) To determine the equations of the tangents to the circle x^2 + y^2 = 20 at A and B, we use the tangent formula x x_1 + y y_1 = r^2. Here, r^2 = 20. Step 1: Find the equation of the tangent at A(-4, -2). x(-4) + y(-2) = 20 -4x - 2y = 20 Divide by -2: 2x + y = -10 Step 2: Find the equation of the tangent at B(2, 4). x(2) + y(4) = 20 2x + 4y = 20 Divide by 2: x + 2y = 10 The equations of the tangents are 2x + y = -10 and x + 2y = 10. f) To determine the coordinates of C, the point of intersection of the tangents, we solve the system of equations from part (e). Step 1: Set up the system of equations. 1) 2x + y = -10 2) x + 2y = 10 Step 2: Solve for x and y. From equation (1), y = -2x - 10. Substitute this into equation (2). x + 2(-2x - 10) = 10 x - 4x - 20 = 10 -3x = 30 x = -10 Step 3: Substitute x = -10 back into y = -2x - 10. y = -2(-10) - 10 y = 20 - 10 y = 10 The coordinates of C are (-10, 10). Drop the next question 📸