Question

A point P(x, 7) divides a line segment joining the points A(-5, 4) and B(7, 9) in a certain ratio. Find the ratio and hence find the value of x.

Hint:

Using $k : 1$ instead of $m_1 : m_2$ for finding ratios is a highly recommended practice because it reduces the number of variables to just one ($k$), making the algebra much simpler and faster to solve!

Answer 1

Step 1: Understanding the Question:
This question is from "Coordinate Geometry".
We are given a point $P(x, 7)$ that lies on the line segment joining points $A(-5, 4)$ and $B(7, 9)$.
Point $P$ divides the segment $AB$ internally in some ratio.
We need to find this division ratio, and then use it to calculate the missing x-coordinate of point $P$.

Step 2: Key Formula or Approach:
We will use the

Section Formula:
If a point $P(x, y)$ divides the line segment joining $A(x_1, y_1)$ and $B(x_2, y_2)$ internally in the ratio $k : 1$, then:
\[ x = \frac{k x_2 + x_1}{k + 1} \] \[ y = \frac{k y_2 + y_1}{k + 1} \] 1. Use the known y-coordinate ($y = 7$) to solve for $k$.
2. Once $k$ is found, substitute it into the x-coordinate formula to find $x$.

Step 3: Detailed Explanation:

• Identify the given coordinate elements:
- $A(x_1, y_1) = (-5, 4)$
- $B(x_2, y_2) = (7, 9)$
- $P(x, y) = (x, 7)$

• Let the ratio in which $P$ divides $AB$ be $k : 1$.
Apply the section formula for the y-coordinate:
\[ y = \frac{k y_2 + y_1}{k + 1} \] \[ 7 = \frac{k(9) + 4}{k + 1} \]

• Solve this equation for $k$:
\[ 7(k + 1) = 9k + 4 \] \[ 7k + 7 = 9k + 4 \] Rearrange terms:
\[ 7 - 4 = 9k - 7k \] \[ 3 = 2k \implies k = \frac{3}{2} \] So the ratio is $\frac{3}{2} : 1$, which is equivalent to $3 : 2$.

• Now, use the section formula for the x-coordinate with the ratio $3 : 2$ (where $m_1 = 3$ and $m_2 = 2$):
\[ x = \frac{m_1 x_2 + m_2 x_1}{m_1 + m_2} \] \[ x = \frac{3(7) + 2(-5)}{3 + 2} \]

• Simplify the numerator and denominator:
\[ x = \frac{21 - 10}{5} \] \[ x = \frac{11}{5} = 2.2 \]

Step 4: Final Answer:
The ratio in which $P$ divides $AB$ is $3 : 2$, and the value of $x$ is $2.2$ (or $\frac{11}{5}$).