Answer:
To find the coordinates of P so that P partitions segment AB in the part-to-whole ratio of 1 to 5 with A(-9, 3) and B(1, 8), we can use the formula:
P = (1 - t)A + tB
where t is a scalar value that represents the position of P on the segment AB, and A and B are the coordinates of the two endpoints of the segment.
In this case, we know that the part-to-whole ratio is 1:5, so t = 1/6.
So we substitute the values for A, B and t into the formula:
P = (1 - 1/6)A + (1/6)B
P = (5/6)A + (1/6)B
P = (-95/6, 35/6) + (1/6, 8/6)
P = (-15/2, 5/2) + (1/6, 4/3)
P = (-15/2 + 1/6, 5/2 + 4/3)
P = (-30/3, 23/6)
The coordinates of P are (-10,3.83)
For the second problem,
To find the coordinates of P so that P partitions the segment AB in the ratio 3 to 1 if A(7, -4) and B(-5, 4), we can use the same formula as above:
P = (1 - t)A + tB
In this case, we know that the part-to-whole ratio is 3:1, so t = 3/4.
So we substitute the values for A, B and t into the formula:
P = (1 - 3/4)A + (3/4)B
P = (1/4)A + (3/4)B
P = (7/4, -4/4) + (-53/4, 43/4)
P = (7/4 + -15/4, -4/4 + 3)
P = (-8/4, -1/4)
The coordinates of P are (-2,-0.25)
So in the first problem point P is (-10,3.83) and in the second problem point P is (-2,-0.25)