If `bar(a),bar(b)andbar(c)` are the position vectors of the points A,B and C respectively, such that
(1) `3bar(a)+5bar(b)=8bar(c)`, find the ratio in which C divides line segment AB.
(2) `3bar(a)+5bar(b)-8bar(c)=bar(0)`, find the ratio in. which A divides BC.

Answer

To solve the given problem, we will break it down into two parts as specified in the question.

### Part (1): Finding the ratio in which C divides the line segment AB

Given the equation:
\[ 3\bar{a} + 5\bar{b} = 8\bar{c} \]

We can rearrange this equation to express \(\bar{c}\):
\[ \bar{c} = \frac{3\bar{a} + 5\bar{b}}{8} \]

Now, we can interpret this in terms of the section formula. The point C divides the line segment AB in the ratio of the coefficients of \(\bar{a}\) and \(\bar{b}\). Here, the coefficients are 3 (for \(\bar{a}\)) and 5 (for \(\bar{b}\)).

Thus, the ratio in which C divides AB is:
\[ \text{Ratio of } C \text{ dividing } AB = 5:3 \]

### Part (2): Finding the ratio in which A divides BC

Given the equation:
\[ 3\bar{a} + 5\bar{b} - 8\bar{c} = \bar{0} \]

This can be rearranged to express \(\bar{a}\):
\[ 3\bar{a} = 8\bar{c} - 5\bar{b} \]
\[ \bar{a} = \frac{8\bar{c} - 5\bar{b}}{3} \]

Now, we can interpret this using the section formula again. Here, A divides the line segment BC in the ratio of the coefficients of \(\bar{b}\) and \(\bar{c}\). The coefficients are -5 (for \(\bar{b}\)) and 8 (for \(\bar{c}\)).

Since A divides BC externally, we take the absolute values of the coefficients:
\[ \text{Ratio of } A \text{ dividing } BC = 8:5 \]

### Final Answers:
1. The ratio in which C divides AB is \(5:3\).
2. The ratio in which A divides BC is \(8:5\).