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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Here are the solutions to your geometry questions. Question 1.1 1.1.1 Determine the relationship between line segments CD and AB. Step 1: Identify the given information. AB is a radius of the circle, and CD is a tangent to the circle at point B. Step 2: Recall the geometric property that a radius drawn to the point of tangency is perpendicular to the tangent. The relationship is that AB is perpendicular to CD. AB CD 1.1.2 Determine the equation of the tangent CD. Step 1: Find the gradient of the line segment AB. The equation of AB is -x + 2y - 1 = 0. Rearrange to y = mx + c form: 2y = x + 1 y = (1)/(2)x + (1)/(2) The gradient of AB is m_AB = (1)/(2). Step 2: Determine the gradient of the tangent CD. Since CD is perpendicular to AB, the product of their gradients is -1. m_CD · m_AB = -1 m_CD · (1)/(2) = -1 m_CD = -2 Step 3: Use the point C (-6, 0) (where CD cuts the x-axis at -6) and the gradient m_CD = -2 to find the equation of CD using the point-slope form y - y_1 = m(x - x_1). y - 0 = -2(x - (-6)) y = -2(x + 6) y = -2x - 12 The equation of the tangent CD is y = -2x - 12. 1.1.3 Determine the coordinates of point B. Step 1: Point B is the intersection of line AB and line CD. We have their equations: Equation of AB: -x + 2y - 1 = 0 (1) Equation of CD: y = -2x - 12 (2) Step 2: Substitute equation (2) into equation (1) to solve for x. -x + 2(-2x - 12) - 1 = 0 -x - 4x - 24 - 1 = 0 -5x - 25 = 0 -5x = 25 x = -5 Step 3: Substitute x = -5 into equation (2) to solve for y. y = -2(-5) - 12 y = 10 - 12 y = -2 The coordinates of point B are (-5, -2). 1.1.4 Determine the equation of the circle in the form (x - a)^2 + (y - b)^2 = r^2. Step 1: Identify the center of the circle. The center is A (-3, -1). So, a = -3 and b = -1. Step 2: Calculate the radius r. The radius is the distance between the center A (-3, -1) and point B (-5, -2) (a point on the circle). Use the distance formula r = sqrt((x_2 - x_1)^2 + (y_2 - y_1)^2). r = sqrt((-5 - (-3))^2 + (-2 - (-1))^2) r = sqrt((-5 + 3)^2 + (-2 + 1)^2) r = sqrt((-2)^2 + (-1)^2) r = sqrt(4 + 1) r = sqrt(5) Step 3: Square the radius to get r^2. r^2 = (sqrt(5))^2 = 5 Step 4: Substitute the center coordinates and r^2 into the standard equation of a circle. (x - (-3))^2 + (y - (-1))^2 = 5 (x + 3)^2 + (y + 1)^2 = 5 The equation of the circle is (x + 3)^2 + (y + 1)^2 = 5. That's 2 down. 3 left today — send the next one.