In what ratio does the point T (2, 0) divide the segment joining the points S (4, -2) and U(1,4) ?

Options

A. ` 2 : 1 `
B. `1 : 2 `
C. ` 2 : 3 `
D. ` 3 : 2 `

Answer

To find the ratio in which the point T (2, 0) divides the segment joining the points S (4, -2) and U (1, 4), we can use the section formula. The section formula states that if a point divides the line segment joining two points (x1, y1) and (x2, y2) in the ratio m:n, then the coordinates of the dividing point can be given by:

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

In our case, we have:
- Point S (x1, y1) = (4, -2)
- Point U (x2, y2) = (1, 4)
- Point T (2, 0) divides the segment.

Let the ratio in which T divides the segment be m:n.

### Step 1: Set up the equations using the section formula
Using the section formula, we can set up the following equations for the x and y coordinates:

For the x-coordinate:
\[ \frac{m \cdot 1 + n \cdot 4}{m+n} = 2 \]

For the y-coordinate:
\[ \frac{m \cdot 4 + n \cdot (-2)}{m+n} = 0 \]

### Step 2: Solve the x-coordinate equation
From the x-coordinate equation:
\[ m + 4n = 2(m + n) \]
Expanding this gives:
\[ m + 4n = 2m + 2n \]
Rearranging terms:
\[ 4n - 2n = 2m - m \]
This simplifies to:
\[ 2n = m \quad \text{(Equation 1)} \]

### Step 3: Solve the y-coordinate equation
From the y-coordinate equation:
\[ 4m - 2n = 0 \]
This can be rearranged to:
\[ 4m = 2n \]
Dividing both sides by 2 gives:
\[ 2m = n \quad \text{(Equation 2)} \]

### Step 4: Substitute Equation 1 into Equation 2
Now we can substitute Equation 1 into Equation 2:
From Equation 1, we have \( m = 2n \). Substituting this into Equation 2:
\[ 2(2n) = n \]
This simplifies to:
\[ 4n = n \]
Subtracting n from both sides gives:
\[ 3n = 0 \implies n = 0 \]
Since \( n = 0 \) is not valid in our ratio, we can use the relationship \( m = 2n \) to express the ratio.

### Step 5: Find the ratio
From \( m = 2n \), we can express the ratio \( m:n \) as:
\[ m:n = 2n:n = 2:1 \]

Thus, the point T (2, 0) divides the segment joining S (4, -2) and U (1, 4) in the ratio **2:1**.