In what ratio does the point T(x, 0) divide the segment joining the points S (-4, -1) and U (1,4) ?

Options

A. `1 : 4`
B. ` 4 : 1 `
C. ` 1 : 2 `
D. ` 2 , 1 `

Answer

To solve the problem of finding the ratio in which the point \( T(x, 0) \) divides the segment joining the points \( S(-4, -1) \) and \( U(1, 4) \), we can use the section formula. Here's a step-by-step solution:

### Step 1: Identify the coordinates
The coordinates of the points are:
- \( S(-4, -1) \) which means \( x_1 = -4 \) and \( y_1 = -1 \)
- \( U(1, 4) \) which means \( x_2 = 1 \) and \( y_2 = 4 \)
- The point \( T \) has coordinates \( (x, 0) \)

### Step 2: Set up the section formula
According to the section formula, if a point \( T \) divides the line segment joining points \( S(x_1, y_1) \) and \( U(x_2, y_2) \) in the ratio \( k:1 \), the coordinates of point \( T \) can be given by:
\[ T\left( \frac{kx_2 + x_1}{k + 1}, \frac{ky_2 + y_1}{k + 1} \right) \]

### Step 3: Apply the formula to find the y-coordinate
Since the y-coordinate of point \( T \) is 0, we can set up the equation for the y-coordinate:
\[ 0 = \frac{k \cdot 4 + (-1)}{k + 1} \]

### Step 4: Solve for k
To solve for \( k \), we can multiply both sides by \( (k + 1) \) (assuming \( k + 1 \neq 0 \)):
\[ 0 = k \cdot 4 - 1 \]
This simplifies to:
\[ k \cdot 4 = 1 \]
\[ k = \frac{1}{4} \]

### Step 5: Express the ratio
The ratio in which point \( T \) divides the segment \( SU \) is \( k:1 \), which is:
\[ \frac{1}{4}:1 \]
This can be expressed as:
\[ 1:4 \]

### Conclusion
Thus, the point \( T(x, 0) \) divides the segment joining points \( S(-4, -1) \) and \( U(1, 4) \) in the ratio \( 1:4 \).

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