Question

The point P(-4, 6) divides the line segment joining A(-6, 10) and B(3, -8) internally in the ratio

• A. 2 : 7
• B. 7 : 2
• C. 3 : 7
• D. 2 : 5

Hint:

Using only one coordinate (either x or y) is sufficient to find the ratio.
Using the x-coordinate is often simpler as the numbers are smaller and less prone to calculation errors.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
This question is from the topic of Coordinate Geometry, specifically involving the division of a line segment.
We are given the coordinates of the endpoints \(A\) and \(B\), and the coordinates of the point of division \(P\).
We need to find the ratio in which \(P\) divides \(AB\).

Step 2: Key Formula or Approach:
According to 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 \(m_1 : m_2\), then:
\[ x = \frac{m_1 x_2 + m_2 x_1}{m_1 + m_2} \]
Let us assume the ratio is \(k : 1\) (where \(k = \frac{m_1}{m_2}\)). The formula simplifies to:
\[ x = \frac{k x_2 + x_1}{k + 1} \]

Step 3: Detailed Explanation: Here, the coordinates are:
\[ A(x_1, y_1) = (-6, 10) \]
\[ B(x_2, y_2) = (3, -8) \]
\[ P(x, y) = (-4, 6) \]
Let the division ratio be \(k : 1\).
Now, apply the Section Formula for the x-coordinate of P:
\[ -4 = \frac{k(3) + 1(-6)}{k + 1} \]
\[ -4(k + 1) = 3k - 6 \]
\[ -4k - 4 = 3k - 6 \]
Rearrange terms to isolate \(k\):
\[ -4 - (-6) = 3k - (-4k) \]
\[ 2 = 7k \]
\[ k = \frac{2}{7} \]
Since the ratio is \(k : 1\), it is:
\[ \frac{2}{7} : 1 \implies 2 : 7 \]
We can verify this ratio using the y-coordinate:
\[ y = \frac{m_1 y_2 + m_2 y_1}{m_1 + m_2} \]
Substitute \(m_1 = 2, m_2 = 7, y_1 = 10, y_2 = -8\):
\[ y = \frac{2(-8) + 7(10)}{2 + 7} = \frac{-16 + 70}{9} = \frac{54}{9} = 6 \]
This matches the y-coordinate of P perfectly, confirming our calculation.

Step 4: Final Answer:
The point P divides the line segment internally in the ratio \(2 : 7\).