To find the point $P$ that partitions the line segment $AB$ in the ratio $2:1$, we use the section formula. The section formula for internal division is given by:

$P(x, y) = \left( \frac{m_2 \cdot x_1 + m_1 \cdot x_2}{m_1 + m_2}, \frac{m_2 \cdot y_1 + m_1 \cdot y_2}{m_1 + m_2} \right)$

Here:

• $A(x_1, y_1) = (0, -3)$
• $B(x_2, y_2) = (0, 4)$
• The ratio $m_1 : m_2 = 2:1$, so $m_1 = 2$ and $m_2 = 1$.

Now applying the section formula:

$P(x, y) = \left( \frac{1 \cdot 0 + 2 \cdot 0}{2+1}, \frac{1 \cdot (-3) + 2 \cdot 4}{2+1} \right)$

Simplifying each coordinate:

$P(x, y) = \left( 0, \frac{-3 + 8}{3} \right)$

$P(x, y) = \left( 0, \frac{5}{3} \right)$

$P(x, y) = \left( 0, 1.67 \right)$

Thus, the point $P$ that partitions the line segment $AB$ in the ratio $2:1$ is $P(0, 1.67)$.

Would you like further details or have any questions?

Here are some related questions:

1. How do you find the midpoint of a line segment?
2. How do you find a point that partitions a line segment externally?
3. How can the section formula be applied in three dimensions?
4. How do you determine the distance between two points on a coordinate plane?
5. What is the significance of a 1:1 partition ratio?

Tip: Always double-check the ratio being used in the section formula to ensure correct application!