Find the ratio in which the point (2, y) divides the join of (- 4, 3) and (6, 3) and hence find the value of y
(a)2 : 3, y = 3
(b)3 : 2, y = 4
(c) 3 : 2, y = 3
(d)3 : 2, y = 2

Options

A. 2 : 3, y = 3
B. 3 : 2, y = 4
C. 3 : 2, y = 3
D. 3 : 2, y = 2

Answer

To solve the problem of finding the ratio in which the point (2, y) divides the line segment joining the points (-4, 3) and (6, 3), we will use the section formula. The section formula states that if a point divides the line segment joining two points \(A(x_1, y_1)\) and \(B(x_2, y_2)\) in the ratio \(m:n\), then the coordinates of the point can be given by:

\[ \left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n} \right) \]

### Step 1: Identify the coordinates and set up the equation
Let the points be:
- \(A(-4, 3)\) where \(x_1 = -4\) and \(y_1 = 3\)
- \(B(6, 3)\) where \(x_2 = 6\) and \(y_2 = 3\)

Let the ratio in which the point (2, y) divides the segment be \(k:1\). Therefore, we have:
- \(m = k\)
- \(n = 1\)

### Step 2: Set up the equation for the x-coordinate
Using the section formula for the x-coordinate:
\[ 2 = \frac{k \cdot 6 + 1 \cdot (-4)}{k + 1} \]

### Step 3: Solve for k
Multiply both sides by \(k + 1\):
\[ 2(k + 1) = 6k - 4 \]
Expanding this gives:
\[ 2k + 2 = 6k - 4 \]
Rearranging the equation:
\[ 2 + 4 = 6k - 2k \]
\[ 6 = 4k \]
\[ k = \frac{6}{4} = \frac{3}{2} \]

### Step 4: Set up the equation for the y-coordinate
Now, we will find the y-coordinate using the same ratio \(k:1\):
\[ y = \frac{k \cdot 3 + 1 \cdot 3}{k + 1} \]
Substituting \(k = \frac{3}{2}\):
\[ y = \frac{\frac{3}{2} \cdot 3 + 1 \cdot 3}{\frac{3}{2} + 1} \]
Calculating the numerator:
\[ y = \frac{\frac{9}{2} + 3}{\frac{3}{2} + 1} = \frac{\frac{9}{2} + \frac{6}{2}}{\frac{3}{2} + \frac{2}{2}} = \frac{\frac{15}{2}}{\frac{5}{2}} = \frac{15}{2} \cdot \frac{2}{5} = 6 \]

### Conclusion
Thus, the point (2, y) divides the line segment in the ratio \(3:2\) and the value of \(y\) is \(3\).

### Final Answer
The ratio is \(3:2\) and \(y = 3\), which corresponds to option (c).

---