Find the ratio in which the line x – $3y = 0$ divides the line segment joining the points (-2, -5) and (6, 3). Find the coordinates of the point of intersection.

Answer: The line $x - 3y = 0$ divides the line segment in the ratio $13:3$ . The coordinates of the point of intersection are $\left( \frac{9}{2}, \frac{3}{2} \right)$ .

Step-by-step solution

Step 1: Assume the ratio and coordinates of intersection point

Let the line $x - 3y = 0$ divide the line segment joining points $A(-2, -5)$ and $B(6, 3)$ in the ratio $k:1$ . We use the section formula to find the coordinates of the point of intersection $P(x, y)$ . Here, $(x_1, y_1) = (-2, -5)$ and $(x_2, y_2) = (6, 3)$ .

Step 2: Apply the section formula

Substitute the coordinates of points A and B into the section formula. This gives us expressions for the $x$ and $y$ coordinates of the intersection point P in terms of $k$ .

Step 3: Substitute into the line equation

Since the point $P(x, y)$ lies on the line $x - 3y = 0$ , its coordinates must satisfy the equation of the line. Substitute the expressions for $x$ and $y$ into the line equation.

Step 4: Solve for k

Simplify the equation by multiplying both sides by $(k+1)$ and then solve for $k$ . This value of $k$ represents the ratio in which the line divides the segment.

Step 5: State the ratio

The value of $k$ we found is $13/3$ . Therefore, the ratio in which the line divides the line segment is $13:3$ .

Step 6: Find the coordinates of the intersection point

Substitute the value of $k = 13/3$ back into the expressions for $x$ and $y$ to find the coordinates of the point of intersection $P(x, y)$ .

Step 7: State the coordinates of the intersection point

The coordinates of the point of intersection are $(9/2, 3/2)$ .

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