Question

Find the ratio in which the x-axis divides the line segment joining the points \((-6, 5)\) and \((-4, -1)\). Also, find the point of intersection.

Hint:

Always assume the ratio as \(k:1\) instead of \(m:n\). It reduces the number of variables and makes the algebra much simpler.

Answer 1

Step 1: Understanding the Concept:
When a line segment is divided by the x-axis, the y-coordinate of the point of intersection is always 0. We use the Section Formula to find the ratio and the specific coordinates.
Step 2: Key Formula or Approach:
Section Formula: \(P(x, y) = \left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n} \right)\)
Step 3: Detailed Explanation:
1. Let the ratio be \(k : 1\). Points are \(A(-6, 5)\) and \(B(-4, -1)\). 2. The y-coordinate of the point of intersection on x-axis is 0: \[ \frac{k(-1) + 1(5)}{k + 1} = 0 \] 3. Solving for \(k\): \(-k + 5 = 0 \implies k = 5\). So, the ratio is 5 : 1. 4. Now find the x-coordinate of the intersection point: \[ x = \frac{5(-4) + 1(-6)}{5 + 1} = \frac{-20 - 6}{6} = \frac{-26}{6} = \frac{-13}{3} \]
Step 4: Final Answer:
The ratio is 5 : 1 and the point of intersection is \(\left(-\frac{13}{3}, 0\right)\).