Problem 89 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If the line \(2 x+y=k\) passes through the point which divides the line segment joining the points \((1,1)\) and \((2,4)\) in the ratio \(3: 2\), then \(k\) equals (A) \(\frac{29}{5}\) (B) 5 (C) 6 (D) \(\frac{11}{5}\)

Step-by-Step Solution

Answer

The value of \( k \) is 6, which is option (C).

Step 1: Determine the Coordinates of the Dividing Point

To find the coordinates of the point that divides the line segment joining points \( (1, 1) \) and \( (2, 4) \) in the ratio \( 3:2 \), use the section formula. The formula for the point dividing the line segment in ratio \( m:n \) is \( \left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n} \right) \). Substituting \( (x_1, y_1) = (1, 1) \) and \( (x_2, y_2) = (2, 4) \) with \( m = 3 \) and \( n = 2 \), we get the coordinates as:

Step 2: Calculate the x-coordinate of the Dividing Point

Using the formula \((3 \cdot 2 + 2 \cdot 1)/(3+2)\), calculate the x-coordinate: \[ x = \frac{3 \times 2 + 2 \times 1}{3 + 2} = \frac{6 + 2}{5} = \frac{8}{5} \]

Step 3: Calculate the y-coordinate of the Dividing Point

Using the formula \((3 \cdot 4 + 2 \cdot 1)/(3+2)\), calculate the y-coordinate: \[ y = \frac{3 \times 4 + 2 \times 1}{3 + 2} = \frac{12 + 2}{5} = \frac{14}{5} \]

Step 4: Substitute Coordinates into the Line Equation

Substitute the coordinates \(\left( \frac{8}{5}, \frac{14}{5} \right)\) into the line equation \(2x + y = k\) to determine \(k\): \[2 \cdot \frac{8}{5} + \frac{14}{5} = k\]

Step 5: Solve for k

Calculate \(k\): \[\frac{16}{5} + \frac{14}{5} = k\]Add the fractions: \[\frac{16+14}{5} = \frac{30}{5} = 6\]

Step 6: Verify the Solution

Verify that the calculated value \(k = 6\) is identical to the option provided. Therefore, option (C) 6 is correct.

Key Concepts

Section Formula

In coordinate geometry, the section formula is a powerful tool to find a specific point on a line segment divided in a particular ratio. Suppose you have two fixed points,

• \((x_1, y_1)\) and \((x_2, y_2)\),
• want to find the point that divides this segment in a given ratio \(m:n\).

The section formula allows you to calculate this point's coordinates systematically. The formula is specified as:

• \( \left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n} \right) \)

This formula effectively finds a weighted average of the endpoints, producing the exact point along the line determined by the ratio.
In the original exercise, we used the formula to find the dividing point of the segment joining points \((1, 1)\) and \((2, 4)\) in the ratio \(3:2\). Calculating both the x and y coordinates, the point came out to be \( \left( \frac{8}{5}, \frac{14}{5} \right) \). This dividing point is crucial to the rest of the problem-solving process.

Line Equation

In coordinate geometry, a line equation represents a straight line and is essential for understanding how different points relate on a plane. The general form of a line equation can be given as:

• \(ax + by + c = 0\), where \(a\), \(b\), and \(c\) are constants.

For any line equation, you can determine various properties of the line, such as slope and intercepts.
In the exercise, the line is given by the equation \(2x + y = k\). This particular line includes all the points \((x, y)\) that satisfy the equation for a specific \(k\) value, indicating its position relative to other geometric entities.
By substituting the point \( \left( \frac{8}{5}, \frac{14}{5} \right) \), which was determined through the section formula, we aimed to find the specific \(k\) for which this point lies on our line. The substituted value of \(k\) ultimately calculated to be 6, aligning with one of the provided answers and ensuring our understanding's accuracy.

Ratio Division

The concept of dividing a line segment into a given ratio is an important aspect of geometrical and algebraic studies. Understanding ratio division helps describe how a point or a line can be partitioned into different parts by this ratio.
If a point divides a line segment in the ratio \(m:n\), it means the segment is divided into two parts whose lengths compare as \(m\) is to \(n\). It's a way of expressing and visualizing a segment being divided into unequal portions according to the given specification.
In the exercise context, we used the ratio \(3:2\) to divide the line segment connecting \((1, 1)\) and \((2, 4)\). This analogy effectively means that the distance from \((1, 1)\) to the dividing point is of a different proportion compared to the rest that continues to \((2, 4)\). Through understanding this division, we were able to utilize the section formula to find accurate coordinates and subsequently determine the value of \(k\) from the line equation, showing the practical application of ratio division in coordinate geometry problems.