Final answer:
Using the section formula in coordinate geometry, none of the provided options exactly match the calculated coordinates for point B, which are (2.2, 0.2). The closest option is (2, 1), suggesting either a rounding scenario or a discrepancy in the provided options.
Step-by-step explanation:
To find the coordinates of point B on line AC such that the ratio of AB to BC is 2:3, we need to use the concept of section formula in coordinate geometry. The section formula tells us how to find the coordinates of a point that divides a line segment into a given ratio.
Let point B have coordinates (x, y). Since the ratio of AB to BC is 2:3, we can express the coordinates of point B using the section formula:
x = (mx₂ + nx₁) / (m + n)
y = (my₂ + ny₁) / (m + n)
Here, (x₁, y₁) = (-1, -1) are the coordinates of point A, (x₂, y₂) = (7, 2) are the coordinates of point C, m = 2, and n = 3. Plugging these values into the section formula, we get:
x = (2×7 + 3×(-1)) / (2 + 3) = (14 - 3) / 5 = 11 / 5 = 2.2
y = (2×2 + 3×(-1)) / (2 + 3) = (4 - 3) / 5 = 1 / 5 = 0.2
Therefore, the coordinates of point B are (2.2, 0.2). Among the given options, none match these exact coordinates, so the correct answer must be one of the options changed to the nearest whole number or a typo occurred with the provided options or in the calculation. The closest option to (2.2, 0.2) is (2, 1), option b).