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.
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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, (x1,y1)=(−2,−5) and (x2,y2)=(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).