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.
ScanSolve AI Answer
You're on a roll — 4.4 Determine the equation of k in the form of k(x) = ax^2 if B is equidistant from A to C. Step 1: Use the equidistant property to find a relationship between the coordinates of B. Let B be (x_B, y_B). A is (6,0) and C is (0,-6). Since B is equidistant from A and C, BA^2 = BC^2. (x_B - 6)^2 + (y_B - 0)^2 = (x_B - 0)^2 + (y_B - (-6))^2 x_B^2 - 12x_B + 36 + y_B^2 = x_B^2 + y_B^2 + 12y_B + 36 -12x_B = 12y_B y_B = -x_B Step 2: Use the fact that B lies on line p to find its coordinates. From 4.3, the equation of line p is y = x - 6. Substitute y_B = -x_B into the line equation: -x_B = x_B - 6 -2x_B = -6 x_B = 3 Now find y_B: y_B = -x_B = -3 So, the coordinates of B are (3, -3). Step 3: Use the coordinates of B to find the value of 'a' for k(x) = ax^2. Point B(3, -3) lies on the parabola k(x) = ax^2. Substitute x=3 and k(x)=-3: -3 = a(3)^2 -3 = 9a a = (-3)/(9) a = -(1)/(3) Step 4: Write the equation of k. Substitute the value of a back into k(x) = ax^2: k(x) = -(1)/(3)x^2 The equation of k is k(x) = -(1)/(3)x^2. Drop the next question.