A,B,C are three collinear points such that AB=2.5 and the co ordinates of A and C are respectively (3,4) and (11,10) then the co ordinates of the point B are

Options

A. `(5,11/2)`
B. `(5,5/2)`
C. `(1,11/2)`
D. `(1,5/2)`

Answer

To find the coordinates of point B, which is collinear with points A and C, we can follow these steps:

### Step 1: Identify the coordinates of points A and C
Given:
- Coordinates of A = (3, 4)
- Coordinates of C = (11, 10)

### Step 2: Calculate the distance AC using the distance formula
The distance formula between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Substituting the coordinates of A and C:
\[ d_{AC} = \sqrt{(11 - 3)^2 + (10 - 4)^2} \]
Calculating:
\[ d_{AC} = \sqrt{(8)^2 + (6)^2} = \sqrt{64 + 36} = \sqrt{100} = 10 \]

### Step 3: Determine the distances AB and BC
Since points A, B, and C are collinear, we know:
\[ AB + BC = AC \]
Given \(AB = 2.5\), we can find \(BC\):
\[ BC = AC - AB = 10 - 2.5 = 7.5 \]

### Step 4: Use the section formula to find the coordinates of B
The section formula states that if a point B divides the line segment AC in the ratio \(m:n\), then the coordinates of B are given by:
\[ B\left(\frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n}\right) \]
Here, \(m = 1\) (for AB) and \(n = 3\) (for BC), since \(BC = 7.5\) is 3 times \(AB = 2.5\).

Substituting the coordinates of A and C:
- \(x_1 = 3\), \(y_1 = 4\)
- \(x_2 = 11\), \(y_2 = 10\)

Now substituting into the section formula:
\[ B\left(\frac{1 \cdot 11 + 3 \cdot 3}{1 + 3}, \frac{1 \cdot 10 + 3 \cdot 4}{1 + 3}\right) \]
Calculating the x-coordinate:
\[ B_x = \frac{11 + 9}{4} = \frac{20}{4} = 5 \]
Calculating the y-coordinate:
\[ B_y = \frac{10 + 12}{4} = \frac{22}{4} = \frac{11}{2} \]

### Step 5: Final coordinates of point B
Thus, the coordinates of point B are:
\[ B = \left(5, \frac{11}{2}\right) \]