Question

Determine the ratio in which \(y-x+2=0\) divides the line joining \((3,-1)\) and \((8,9)\) : (a) \(3: 5\) (b) \(4: 3\) (c) \(2: 3\) (d) none of these

Short answer

a. 1:2 b. 2:1 c. 3:1 d. None of these Answer: (d) None of these

Step-by-step solution

Write the section formula for the given line segment

The section formula for a line segment joining points \((x_1, y_1)\) and \((x_2, y_2)\), divided in the ratio \(m:n\), is given by: $$(x, y) = \left(\frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n}\right)$$ Here, \((x_1, y_1) = (3, -1)\) and \((x_2, y_2) = (8, 9)\). Let the ratio in which the line \(y-x+2=0\) divides the line segment be \(m:n\). So, using the section formula, $$(x, y) = \left(\frac{8m + 3n}{m+n}, \frac{9m - n}{m+n}\right)$$

Find the coordinates of the point that satisfies the given line equation

We know that the point \((x, y)\) on the line segment also lies on the line \(y-x+2=0\). Substitute the \(x\) and \(y\) coordinates from the section formula into the line equation, $$\frac{9m - n}{m+n} - \frac{8m + 3n}{m+n} + 2 = 0$$

Solve the equation to find the ratio \(m:n\)

Multiply both sides of the equation by \((m + n)\) to eliminate the denominators, $$(9m - n) - (8m + 3n) + 2(m + n) = 0$$ Simplify the equation, $$m - 5n + 2m + 2n = 0$$ Combine like terms, $$3m - 3n = 0$$ Divide both sides by 3, $$m - n = 0$$ This implies that \(m = n\). Let's assume that \(m = n = 1\). Hence the ratio in which the given line divides the line segment is \(1:1\). This is not among the given options, so the correct answer is (d) none of these.

Key concepts

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is a branch of mathematics that allows us to analyze geometric shapes through a system of coordinates. It is a powerful tool that translates geometric problems into algebraic equations using a coordinate plane, which consists of a horizontal axis (x-axis) and a vertical axis (y-axis).

Every point in the plane can be represented by a pair of numbers \( (x, y) \) which are its coordinates: \( x \) denotes its horizontal distance from the origin, and \( y \) its vertical distance. The intersection of the horizontal and vertical axes at the origin (0, 0) enables us to position every possible point using these two references. This fundamental concept is the foundation of many geometric algorithms, including the section formula, which provides a method to find a point that divides a line segment into a particular ratio.

Ratio and Proportion

Ratio and proportion are fundamental concepts in mathematics that describe the relationship between quantities. A ratio is a comparison of two quantities showing the number of times one value contains or is contained within the other, typically represented as \( a:b \) or \( \frac{a}{b} \).

In the context of coordinate geometry, ratios help us understand how a line segment is divided. If a point divides a line segment in the ratio \( m:n \) this means that the distance from the point to one end of the segment is \( m \) times some unit, while the distance to the other end is \( n \) times that same unit. Proportion, meanwhile, represents the equality of two ratios, which is crucial when solving problems related to the division of line segments. For example, if two line segments are proportional, their corresponding lengths are in the same ratio.

Line Segment Division

The concept of line segment division involves finding a point that divides a line segment into two parts with a specific ratio. Using the section formula, we can determine the coordinates of this dividing point.

The section formula is an expression used in coordinate geometry that allows us to find a point that divides a line segment into a given ratio \( m:n \) by applying the weights \( m \) and \( n \) to the coordinates of the endpoints of the segment. In the exercise provided, we saw how the formula was applied to the endpoints of the segment to ascertain the point of division. This process is not only used in pure mathematics but also in various practical fields such as computer graphics and engineering to calculate intermediate points in design and modeling.