Understanding the Section Formula for Internal Division

The problem asks us to find the coordinates of point B, given that point C divides the line segment joining A and B internally in a specific ratio. This type of problem is solved using the section formula in coordinate geometry.

The section formula states that if a point \( C(x, y) \) divides the line segment joining \( A(x_1, y_1) \) and \( B(x_2, y_2) \) internally in the ratio \( m : n \), then the coordinates of \( C \) are given by:

\( x = \frac{mx_2 + nx_1}{m+n} \)

\( y = \frac{my_2 + ny_1}{m+n} \)

Applying the Section Formula to Find Coordinates of B

We are given the following information:

• Point A: \( (x_1, y_1) = (-2, 7) \)
• Point C: \( (x, y) = (1, 1) \)
• Ratio \( m : n = 3 : 2 \) (C divides AB in the ratio 3:2)
• Point B: \( (x_2, y_2) = (x_B, y_B) \) (unknown coordinates we need to find)

Let's use the section formula to find the x-coordinate of B, \( x_B \).

\( x = \frac{mx_2 + nx_1}{m+n} \)

Substitute the known values:

\( 1 = \frac{3 \cdot x_B + 2 \cdot (-2)}{3+2} \)

\( 1 = \frac{3x_B - 4}{5} \)

Multiply both sides by 5:

\( 1 \times 5 = 3x_B - 4 \)

\( 5 = 3x_B - 4 \)

Add 4 to both sides:

\( 5 + 4 = 3x_B \)

\( 9 = 3x_B \)

Divide by 3:

\( x_B = \frac{9}{3} \)

\( x_B = 3 \)

Now, let's use the section formula to find the y-coordinate of B, \( y_B \).

\( y = \frac{my_2 + ny_1}{m+n} \)

Substitute the known values:

\( 1 = \frac{3 \cdot y_B + 2 \cdot 7}{3+2} \)

\( 1 = \frac{3y_B + 14}{5} \)

Multiply both sides by 5:

\( 1 \times 5 = 3y_B + 14 \)

\( 5 = 3y_B + 14 \)

Subtract 14 from both sides:

\( 5 - 14 = 3y_B \)

\( -9 = 3y_B \)

Divide by 3:

\( y_B = \frac{-9}{3} \)

\( y_B = -3 \)

So, the coordinates of point B are \( (3, -3) \).

Verifying the Coordinates of B

The calculated coordinates for point B are \( (3, -3) \). Let's compare this with the given options.

The calculated coordinates \( (3, -3) \) match the coordinates in Option 2.