Assertion: (-3, 10) and (6, -8) are divided in the ratio 2:7 by the point (6, 1).

Reason: Three points a, b, and c are collinear if the ratio for both a and b is the same.

Explanation:

Collinearity of Points:
Collinearity refers to the property of points lying on the same straight line. Three points A, B, and C are collinear if they lie on the same line. In other words, the line passing through A and B also passes through C.

Ratio of Division:
When a point divides a line segment into two parts, the ratio of the lengths of the two parts is determined. For example, if a point P divides a line segment AB into two parts in the ratio m:n, then AP:PB = m:n.

Collinearity and Ratio:
If three points are collinear, the ratio of the lengths of the line segments formed by these points will be the same. In other words, if a point P divides AB and CD in the same ratio, then A, B, and C are collinear.

Application of the Reason:
In this case, the two given points (-3, 10) and (6, -8) are divided in the ratio 2:7 by the point (6, 1). Let's calculate the ratios of the line segments formed by these points.

The length of the line segment formed by (-3, 10) and (6, 1) can be calculated using the distance formula:

Length AB = √[(x2 - x1)^2 + (y2 - y1)^2]
= √[(6 - (-3))^2 + (1 - 10)^2]
= √[9^2 + (-9)^2]
= √[81 + 81]
= √162
= 9√2

The length of the line segment formed by (6, 1) and (6, -8) can be calculated as the difference in their y-coordinates:

Length BC = |-8 - 1|
= |-9|
= 9

Now, let's calculate the ratios of these line segments:

Ratio AB:BC = (9√2):9
= √2:1

Since the ratio of the line segments AB and BC is the same, i.e., √2:1, we can conclude that the points (-3, 10), (6, 1), and (6, -8) are collinear.

Hence, the assertion that (-3, 10) and (6, -8) are divided in the ratio 2:7 by the point (6, 1) is correct, and the reason provided is valid.