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
Step 1: To find the points where the locus meets the y-axis, set x=0 in the equation of the locus found in part (a). The equation of the locus is 3x^2 + 3y^2 + 12x - 38y + 55 = 0. Substitute x=0: 3(0)^2 + 3y^2 + 12(0) - 38y + 55 = 0 3y^2 - 38y + 55 = 0 Step 2: Solve the quadratic equation for y using the quadratic formula y = -b ± sqrt(b^2 - 4ac)2a. Here, a=3, b=-38, and c=55. y = -(-38) ± sqrt((-38)^2 - 4(3)(55))2(3) y = 38 ± sqrt(1444 - 660)6 y = 38 ± sqrt(784)6 Step 3: Calculate the square root and find the two values for y. sqrt(784) = 28 y = (38 ± 28)/(6) The two possible values for y are: y_1 = (38 + 28)/(6) = (66)/(6) = 11 y_2 = (38 - 28)/(6) = (10)/(6) = (5)/(3) Step 4: State the points where the locus meets the y-axis. Since x=0, the points are (0, y_1) and (0, y_2). The points are (0, 11) and (0, (5)/(3)). What's next?