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:

A quick formula for the ratio in which \(x\)-axis divides \((x_1, y_1)\) and \((x_2, y_2)\) is \(-y_1 : y_2\).
Here, \(-5 : -1\) which simplifies to \(5 : 1\).

Answer 1

Step 1: Understanding the Concept:
Any point on the \(x\)-axis has a \(y\)-coordinate of 0. Let the ratio be \(k : 1\). We use the section formula to find \(k\).
Step 2: Key Formula or Approach:
Section formula: \(x = \frac{mx_2 + nx_1}{m+n}, y = \frac{my_2 + ny_1}{m+n}\)
Step 3: Detailed Explanation:
Let the ratio be \(k : 1\). The points are \(A(6, 5)\) and \(B(-4, -1)\).
The point of intersection on \(x\)-axis is \((x, 0)\).
Using the \(y\)-coordinate:
\[ 0 = \frac{k(-1) + 1(5)}{k + 1} \]
\[ -k + 5 = 0 \implies k = 5 \]
So the ratio is \(5 : 1\).
Now, find the \(x\)-coordinate:
\[ x = \frac{5(-4) + 1(6)}{5 + 1} = \frac{-20 + 6}{6} = \frac{-14}{6} = -\frac{7}{3} \]
Step 4: Final Answer:
The ratio is \(5 : 1\) and the point of intersection is \((-7/3, 0)\).