What is the value of k if the point (-3, k) divides the line segment joining the points (-5, -4) and (-2, 3) in a certain ratio?

Options

A. `-1`
B. 3
C. 2
D. `(2)/(3)`

Answer

To find the value of \( k \) such that the point \((-3, k)\) divides the line segment joining the points \((-5, -4)\) and \((-2, 3)\) in a certain ratio, we can use the section formula.

### Step 1: Identify the points and the section formula
The points are:
- \( A(-5, -4) \)
- \( B(-2, 3) \)
- \( P(-3, k) \)

The section formula states that if a point \( P(x, y) \) divides the line segment joining points \( A(x_1, y_1) \) and \( B(x_2, y_2) \) in the ratio \( m:n \), then:
\[ x = \frac{mx_2 + nx_1}{m+n} \quad \text{and} \quad y = \frac{my_2 + ny_1}{m+n} \]

### Step 2: Set up the equations for \( x \) and \( y \)
Assuming the ratio is \( 1:\lambda \), we can write:
\[ -3 = \frac{1 \cdot (-2) + \lambda \cdot (-5)}{1 + \lambda} \]
\[ k = \frac{1 \cdot 3 + \lambda \cdot (-4)}{1 + \lambda} \]

### Step 3: Solve for \( \lambda \) using the \( x \)-coordinate
From the first equation:
\[ -3(1 + \lambda) = -2 + (-5\lambda) \]
Expanding this gives:
\[ -3 - 3\lambda = -2 - 5\lambda \]
Rearranging terms:
\[ -3 + 2 = -5\lambda + 3\lambda \]
\[ -1 = -2\lambda \]
Thus,
\[ \lambda = \frac{1}{2} \]

### Step 4: Substitute \( \lambda \) into the \( y \)-coordinate equation
Now substituting \( \lambda = \frac{1}{2} \) into the equation for \( k \):
\[ k = \frac{1 \cdot 3 + \frac{1}{2} \cdot (-4)}{1 + \frac{1}{2}} \]
Calculating the numerator:
\[ k = \frac{3 - 2}{1 + \frac{1}{2}} = \frac{1}{\frac{3}{2}} = \frac{1 \cdot 2}{3} = \frac{2}{3} \]

### Conclusion
Thus, the value of \( k \) is:
\[ \boxed{\frac{2}{3}} \]