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: Find the parametric equation of line (AB). The coordinates of points A and B are A(1,0) and B(-2,3). First, find the direction vector AB. AB = (x_B - x_A, y_B - y_A) = (-2 - 1, 3 - 0) = (-3, 3) The parametric equation of a line passing through a point (x_0, y_0) with a direction vector (u,v) is given by: x = x_0 + t · u \\ y = y_0 + t · v Using point A(1,0) and the direction vector AB=(-3,3), and letting the parameter be t: x = 1 + t(-3) \\ y = 0 + t(3) x = 1 - 3t \\ y = 3t The student used m as the parameter, which is also acceptable. (AB): x = 1 - 3m \\ y = 3m Step 2: Find the Cartesian equation of line (d). The line (d) is given by the parametric equations: x = m+1 (1) \\ y = -2m+3 (2) From equation (1), express m in terms of x: m = x-1 Substitute this expression for m into equation (2): y = -2(x-1)+3 y = -2x+2+3 y = -2x+5 To write the equation in the standard Cartesian form ax+by+c=0, move all terms to one side: 2x+y-5 = 0 The student's work for this part is correct. (d): 2x+y-5=0