What are the coordinates of the point which divides the line joining 5 -2 and 9 6 internally in the ratio 1:3?

The point that divides a line segment joining two points in a given ratio is a common problem in coordinate geometry. In this article, we will solve such a problem step-by-step using the section formula. We will also discuss how to apply this formula generally and explore its importance in various mathematical contexts.

Section Formula Overview

The section formula is a useful tool in coordinate geometry, particularly for finding the coordinates of a point that divides a line segment joining two points in a given internal ratio. For points P(x1, y1) and Q(x2, y2), if a point R(x, y) divides the line segment PQ in the ratio m:n, then the coordinates of the point R(x, y) can be calculated using the formula:

R(x, y) left( frac{mx2 nx1}{m n}, frac{my2 ny1}{m n} right)

Application of the Section Formula

Let's solve the problem at hand: finding the coordinates of the point which divides the line segment joining the points (5, -2) and (9, 6) in the ratio 1:3.

Step 1: Identify the Given Points and Ratio

Step 2: Substitute into the Section Formula

Let's find the x-coordinate:

x (frac{1 cdot 9 3 cdot 5}{1 3} frac{9 15}{4} frac{24}{4} 6)

Now, let's find the y-coordinate:

y (frac{1 cdot 6 3 cdot (-2)}{1 3} frac{6 - 6}{4} frac{0}{4} 0)

Step 3: Conclusion

The coordinates of the point that divides the line segment in the ratio 1:3 are ((6, 0)).

General Form of the Section Formula

For a more general approach, the formula can be written as:

The point that divides the line segment joining points ((x1, y1)) and ((x2, y2)) in the ratio (a:b) is given by:

(left( frac{ax2 bx1}{a b}, frac{ay2 by1}{a b} right))

Application to Olympiad Problems

This concept is particularly useful for solving problems in mathematics competitions like the Olympiads. For instance, in the given problem, the point that divides the line segment joining (5, -2) and (9, 6) in the ratio 1:3 is ((6, 0)).

Conclusion

The section formula is a fundamental tool in coordinate geometry, and its application can help us find the coordinates of a point that divides a line segment in a given ratio. Understanding and mastering this concept will greatly assist in solving a variety of mathematical problems, from basic geometry to advanced Olympiad questions.