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 find 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. The section formula states that if a point divides a line segment joining two points in the ratio m:n, then the coordinates of the point can be expressed as:

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

Here, we have:
- Point S(-4, -1) as (x1, y1)
- Point U(1, 4) as (x2, y2)
- Point T(x, 0) as (x, y)

We need to find the ratio \( m:n \) such that the y-coordinate of point T is 0.

### Step 1: Set up the equation for the y-coordinate
Using the section formula for the y-coordinate, we have:

\[ 0 = \frac{m \cdot 4 + n \cdot (-1)}{m+n} \]

### Step 2: Simplify the equation
Multiplying both sides by \( m+n \) to eliminate the denominator gives:

\[ 0 = 4m - n \]

### Step 3: Rearranging the equation
Rearranging the equation gives:

\[ n = 4m \]

### Step 4: Express the ratio
Now, we can express the ratio \( m:n \):

\[ \frac{m}{n} = \frac{m}{4m} = \frac{1}{4} \]

Thus, the ratio \( m:n \) is \( 1:4 \).

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

---