We are given two points, $A(2, -6)$ and $B(-3, 0)$, and we need to find the coordinates of the point $P$ that divides the line segment $AB$ externally in the ratio 7:2. The formula for the coordinates of a point dividing a line externally is:

$P\left( \frac{m_1x_2 - m_2x_1}{m_1 - m_2}, \frac{m_1y_2 - m_2y_1}{m_1 - m_2} \right)$ where:

• $A(x_1, y_1)$ and $B(x_2, y_2)$ are the coordinates of the two points,
• $m_1:m_2$ is the given ratio, and
• It’s external division, so subtraction is used in the denominator.

Step 1: Identify the values

• $x_1 = 2$, $y_1 = -6$ (coordinates of $A$),
• $x_2 = -3$, $y_2 = 0$ (coordinates of $B$),
• $m_1 = 7$, $m_2 = 2$ (given ratio).

Step 2: Apply the formula

For the x-coordinate of point $P$: $x = \frac{7 \times (-3) - 2 \times 2}{7 - 2} = \frac{-21 - 4}{5} = \frac{-25}{5} = -5$

For the y-coordinate of point $P$: $y = \frac{7 \times 0 - 2 \times (-6)}{7 - 2} = \frac{0 + 12}{5} = \frac{12}{5}$

Final answer:

The coordinates of the point $P$ are $P(-5, \frac{12}{5})$, which matches the provided answer in the image.

Would you like more details or have any questions?

Related Questions:

1. How do we calculate the point that divides a line segment internally?
2. What is the difference between internal and external division of a line segment?
3. How do you find the distance between two points using the distance formula?
4. Can we use the section formula in three dimensions as well?
5. How is the section formula derived from vector geometry?

Tip:

When dividing a line externally, always remember to subtract the terms in the denominator.