The point `(10/7,33/7)` divides `P(1, 3)` and `Q(2,7)` internally in the ratio :

Options

A. `3:4`
B. `3:2`
C. `2:3`
D. `4:3`

Answer

To find the ratio in which the point \((\frac{10}{7}, \frac{33}{7})\) divides the line segment joining the points \(P(1, 3)\) and \(Q(2, 7)\) internally, we can use the section formula. The section formula states that if a point \(R(x, y)\) divides the line segment joining the points \(P(x_1, y_1)\) and \(Q(x_2, y_2)\) in the ratio \(m:n\), then:

\[ x = \frac{mx_2 + nx_1}{m+n} \]
\[ y = \frac{my_2 + ny_1}{m+n} \]

### Step-by-Step Solution:

1. **Assign Coordinates**:
- Let \(P(1, 3)\) be \((x_1, y_1)\) and \(Q(2, 7)\) be \((x_2, y_2)\).
- The coordinates of the point \(R\) are given as \((\frac{10}{7}, \frac{33}{7})\).

2. **Set Up the Equations**:
- Using the section formula for the x-coordinates:
\[ \frac{10}{7} = \frac{m \cdot 2 + n \cdot 1}{m + n} \]
- Using the section formula for the y-coordinates:
\[ \frac{33}{7} = \frac{m \cdot 7 + n \cdot 3}{m + n} \]

3. **Cross-Multiply to Eliminate Fractions**:
- For the x-coordinate equation:
\[ 10(m + n) = 14m + 10n \]
Simplifying gives:
\[ 10m + 10n = 14m + 10n \implies 10m = 14m - 10n \implies 4m = 3n \implies \frac{m}{n} = \frac{3}{4} \]
- For the y-coordinate equation:
\[ 33(m + n) = 49m + 21n \]
Simplifying gives:
\[ 33m + 33n = 49m + 21n \implies 33m + 12n = 49m \implies 16m = 12n \implies \frac{m}{n} = \frac{3}{4} \]

4. **Conclude the Ratio**:
- From both equations, we find that the ratio \(m:n\) is \(3:4\).

### Final Answer:
The point \((\frac{10}{7}, \frac{33}{7})\) divides the line segment joining \(P(1, 3)\) and \(Q(2, 7)\) in the ratio \(3:4\).