Final answer:
The possible images of point M(1,2) after a translation along a vector parallel to the line y=2x+4 with a length of √5 are (2,4) and (0,0). This is because the direction of the vector can either be (1,2) or its negative (-1,-2) and the original point must be adjusted accordingly.
Step-by-step explanation:
The question involves finding the possible images of a point after a translation along a vector. The original point is M(1,2) and it is translated along a vector parallel to the line y = 2x + 4. The length of the translation vector is given to be √5 (the square root of 5).
The slope of the line y = 2x + 4 is 2, which can be represented by the direction vector (1,2) since the y-value changes by 2 when the x-value changes by 1 in this slope. The actual translation vector must have the same direction but be scaled to have a length of √5. It is possible to scale the vector (1,2) to obtain the vector with magnitude √5. This can be done by dividing the vector (1,2) by its current length, which is √(1^2+2^2) = √5, and then multiplying it by the desired length √5. However, doing this would yield the vector (1,2), so the scaling step is not necessary in this case since the vector already has the desired length.
Thus, the translation vector can either be (1,2) or its negative (-1,-2) since translations can be in two directions along the same line. To translate point M, we add the translation vector to the coordinates of M to find the new positions:
- Translation using (1,2): New point = (1+1, 2+2) = (2,4)
- Translation using (-1,-2): New point = (1-1, 2-2) = (0,0)
Therefore, the images of point M after the translation could be at (2,4) or at (0,0).