Final answer:
The student's question pertains to finding the equations of the angle bisectors between two lines and proving that they are perpendicular. The problem uses the coefficients of the lines to derive the bisector equations and involves showing the slopes' product is -1. The bisector containing the origin will satisfy the equation when both x and y are zero.
Step-by-step explanation:
To find the equations of the bisectors of the angles between the lines 4x – 3y + 1 = 0 and 12x - 5y + 7 = 0, we should first consider the formulas for finding the components of vectors. However, the concept of vectors and their components is more applicable to physics problems, and our given problem is more geometrical in nature. This seems to be a mix-up in the brief.
To tackle the mathematical problem appropriately, we analyze the given lines and use properties of angle bisectors. Each line can be written in the form Ax + By + C = 0, where A, B, and C are coefficients and x, y are variables. The angle bisectors of two intersecting lines have equations that can be derived using the coefficients of the given lines. Specifically, if the lines have equations A1x + B1y + C1 = 0 and A2x + B2y + C2 = 0, then the angle bisectors are given by A1x + B1y + C1 = ±(A2x + B2y + C2)
After finding these equations, proving that the bisectors are at right angles to each other involves demonstrating that the product of their slopes is -1. Lastly, the bisector that contains the origin can be identified by setting x = 0 and y = 0 in the equation of the angle bisector that yields a true statement.