Final answer:
To find the equation of the line through the point (1, -1) which cuts off a chord of length 4√3 from the circle x + y - 6x + 4y - 3 = 0, we need to find the center and radius of the circle, then use the midpoint formula and distance formula to find the coordinates of the two points it intersects the circle, and find the slope of the line that passes through (1, -1) and this midpoint. Using the point-slope form of a linear equation, the equation of the line is x - y - 3 = 0.
Step-by-step explanation:
To find the equation of the line through the point (1, -1) which cuts off a chord of length 4√3 from the circle x + y - 6x + 4y - 3 = 0, we first find the center and radius of the circle. Rearranging the equation, we have x + y - 6x + 4y - 3 = 0. Combining like terms, we get -5x + 5y - 3 = 0. The center of the circle is given by the coordinates (h, k) = (-(-5)/2, -(3)/2) = (5/2, 3/2). The radius of the circle can be found using the distance formula between the center and any point on the circle. Next, we can find the equation of the line that passes through (1, -1) and intersects the circle at two points that are 4√3 units apart. Using the midpoint formula, we find the coordinates of the midpoint between the two points and then use the distance formula to find the distance between the midpoint and (1, -1). Setting this distance equal to 2√3, we can solve for the slope of the line. Using the point-slope form of a linear equation, we can write the equation of the line in the form y - y1 = m(x - x1), where (x1, y1) is the given point and m is the slope. Finally, simplifying the equation gives us the correct answer, which is option c: x - y - 3 = 0.