Final answer:
The point (3, b) divides the segment joining the points A (7, 1) and B(0,8) in the ratio 7:3 (or 7/3:1), and the value of 'b' is 8.
Step-by-step explanation:
The student is asking to find the ratio in which the point (3, b) divides the segment joining points A (7, 1) and B (0, 8) and to find the value of 'b'. This can be solved using the section formula which is used to find the coordinates of a point which divides a line segment in a given ratio. To find the ratio, we can equate the x-coordinate of the point that divides the segment, which is 3, to the x-coordinate given by the section formula. For a line segment divided by a point (x, y) in the ratio m:n, the section formula gives the x-coordinate as (mx2 + nx1)/(m + n) and the y-coordinate as (my2 + ny1)/(m + n), where (x1, y1) and (x2, y2) are the endpoints of the line segment.
By applying the section formula to the x-coordinate:
- 3 = (m*0 + n*7) / (m + n)
- m = n*7/3 - n
Now, let's find the value of 'b' using the y-coordinate:
- b = (m*8 + n*1) / (m + n)
Substituting m from the previous equation, we get:
- b = ((n*7/3 - n)*8 + n*1) / (n*7/3 - n + n)
- b= (56/3 - 8)n / (4/3)n
- b = 14 - 6 = 8
Thus, the point (3, b) divides the segment AB in the ratio 7:3 (or 7/3:1 to be precise) and the value of 'b' is 8.