The point `(11 ,10)` divides the line segment joining the points (5,-2) and (9,6) in the ratio:

Options

A. (a) `1:3` internally
B. (b) `1: 3` externally
C. (c) `3: 1` internally
D. (d) `3: 1` externally

Answer

To solve the problem of finding the ratio in which the point (11, 10) divides the line segment joining the points (5, -2) and (9, 6), we will use the section formula. Here's a step-by-step solution:

### Step 1: Identify the Points
Let:
- Point A = (5, -2) = (x1, y1)
- Point B = (9, 6) = (x2, y2)
- Point P = (11, 10) = (x, y)

### Step 2: Use the Section Formula
The section formula states that if a point P divides the line segment joining points A and B in the ratio m:n, then:
\[ x = \frac{mx2 + nx1}{m + n} \]
\[ y = \frac{my2 + ny1}{m + n} \]

### Step 3: Set Up the Equations
Assuming the ratio is m:n, we can express the coordinates as:
\[ 11 = \frac{m \cdot 9 + n \cdot 5}{m + n} \quad \text{(1)} \]
\[ 10 = \frac{m \cdot 6 + n \cdot (-2)}{m + n} \quad \text{(2)} \]

### Step 4: Solve for m and n
From equation (1):
Cross-multiplying gives:
\[ 11(m + n) = 9m + 5n \]
Expanding:
\[ 11m + 11n = 9m + 5n \]
Rearranging:
\[ 11m - 9m + 11n - 5n = 0 \]
\[ 2m + 6n = 0 \quad \Rightarrow \quad 2m = -6n \quad \Rightarrow \quad m = -3n \]

### Step 5: Substitute m in terms of n
Let n = 1 (for simplicity):
Then,
\[ m = -3 \cdot 1 = -3 \]

### Step 6: Determine the Ratio
The ratio m:n is:
\[ -3:1 \]
Since m is negative, this indicates that the point divides the segment externally.

### Step 7: Express the Ratio in Positive Terms
The external division can be expressed as:
\[ 3:1 \quad \text{(external)} \]

### Conclusion
Thus, the point (11, 10) divides the line segment joining the points (5, -2) and (9, 6) in the ratio **3:1 externally**.

### Final Answer
The correct option is **D: 3 is to 1 externally**.

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