Question

The ratio in which the line joining points \((a+b, b+a)\) and \((a-b, b-a)\) is divided by the point \((a, b)\) is (1) b: a internally (2) \(1: 1\) internally (3) \(\mathrm{a}: \mathrm{b}\) externally (4) \(2: 1\) externally

Short answer

The ratio in which the point \((a, b)\) divides the line segment joining points \((a+b, b+a)\) and \((a-b, b-a)\) is \(1:1\) internally.

Step-by-step solution

Write Down the Given Points

The coordinates of the three points are: Point P: \((a+b, b+a)\), Point Q: \((a-b, b-a)\), Point R: \((a, b)\).

Apply Section Formula

Let point R \((a, b)\) divide the line segment PQ in the ratio \(k_1:k_2\). According to section formula, the coordinates of point R are given by: $$\left(\frac{k_1x_2 + k_2x_1}{k_1+k_2}, \frac{k_1y_2 + k_2y_1}{k_1+k_2}\right)$$ where, P\((x_1, y_1) = (a+b, b+a)\) and Q\((x_2, y_2) = (a-b, b-a)\). Now we need to find values of \(k_1\) and \(k_2\).

Substitute the Point R and Solve for k_1 and k_2

Plug the coordinates of point R into the section formula and equate it with both coordinates of the point, respectively. $$\frac{k_1(a-b) + k_2(a+b)}{k_1+k_2} = a$$ $$\frac{k_1(b-a) + k_2(b+a)}{k_1+k_2} = b$$ Solving these two equations, we get: $$k_1(a-b) + k_2(a+b) = (k_1+k_2)a$$ $$k_1(b-a) + k_2(b+a) = (k_1+k_2)b$$

Simplify and Compare with Given Options

Simplifying the equations and comparing with the given options, we see: $$k_1a - k_1b + k_2a + k_2b = k_1a + k_2a$$ $$k_1b - k_1a + k_2b + k_2a = k_1b + k_2b$$ Looking at these two identities, we can conclude: $$-k_1b + k_2b = 0$$ $$k_1b = k_2b$$ Dividing both sides by \(b\), we get \(k_1=k_2\). Thus, the ratio of \(k_1 : k_2 = 1:1\) internally. #Answer# The ratio in which the line joining points \((a+b, b+a)\) and \((a-b, b-a)\) is divided by the point \((a, b)\) is \(1:1\) internally (Option 2).

Key concepts

Understanding Coordinate Geometry

Coordinate geometry is a branch of mathematics that involves graphing and dealing with points, lines, and figures using a coordinate system. The most common system used is the Cartesian coordinate system, where each point on a plane is defined by a pair of numerical coordinates. These coordinates represent the point's position in terms of its distance from two fixed perpendicular axes.

In the context of the exercise provided, we're dealing with points in a two-dimensional space where each point has an x-coordinate (horizontal position) and a y-coordinate (vertical position). By knowing the coordinates of points, we can solve a variety of geometrical problems, including finding the ratio in which a line segment is divided by a point, which is precisely what we're addressing in this exercise.

Grasping Ratio and Proportion

Ratio and proportion are foundational concepts in mathematics that describe the relationship between quantities. A ratio is a way to compare two quantities by division, shown typically as 'a to b' or as a fraction \(\frac{a}{b}\). Proportion, on the other hand, is an equation that states two ratios are equal. It is a statement that two ratios (or fractions) are equivalent.

When it comes to the section formula in coordinate geometry, these concepts help determine how a point divides a line segment. If a point divides a segment into two parts in the ratio \(k_1:k_2\), the proportions of the segment lengths to the respective distances from the given point are \(k_1\) and \(k_2\). This idea is central to solving problems like the one presented in the exercise where we need to find the ratio in which a point divides a segment.

Line Segment Division in Detail

The division of a line segment refers to finding a point on the segment that divides it into two parts in a specific ratio. The section formula—which is an essential tool in coordinate geometry—is used to find the coordinates of the point that divides the line segment. \

The section formula states that if a point \(\text{R}(x, y)\) divides the line segment joining points \(\text{P}(x_1, y_1)\) and \(\text{Q}(x_2, y_2)\) in the ratio \(k_1:k_2\), then the coordinates of \(\text{R}\) can be calculated using: \[\left(\frac{k_1x_2 + k_2x_1}{k_1+k_2}, \frac{k_1y_2 + k_2y_1}{k_1+k_2}\right)\] This formula allows us to find the exact position of the dividing point on the line segment, which in our exercise is point \(\text{R}(a, b)\). By plugging in the coordinates of \(\text{R}\) and equating them with the coordinates generated by the section formula, we were able to determine that the segment was divided internally in a ratio of \(1:1\).