Problem 94 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

Let \(P S\) be the median of the triangle with vertices \(P\) \((2,2), Q(6,-1)\) and \(R(7,3)\). The equation of the line passing through \((1,-1)\) and parallel to \(P S\) is (A) \(4 x-7 y-11=0\) (B) \(2 x+9 y+7=0\) (C) \(4 x+7 y+3=0\) (D) \(2 x-9 y-11=0\)

Step-by-Step Solution

Answer

The correct answer is (B) \(2x + 9y + 7 = 0\).

Step 1: Calculate the midpoint of QR

To find the equation of the median, we first need to find the midpoint of side \(QR\). Given the points \(Q(6, -1)\) and \(R(7, 3)\): \[\text{Midpoint} = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)\]Substituting in the values:\[\text{Midpoint} = \left(\frac{6 + 7}{2}, \frac{-1 + 3}{2}\right) = \left(\frac{13}{2}, 1\right)\]

Step 2: Find the slope of median PS

Now, using point \(P(2, 2)\) and the midpoint of \(QR\) \(\left(\frac{13}{2}, 1\right)\), we can calculate the slope of the median \(PS\).The slope \(m\) is given by: \[m = \frac{y_2 - y_1}{x_2 - x_1}\ = \frac{1 - 2}{\frac{13}{2} - 2} = \frac{-1}{\frac{9}{2}} = -\frac{2}{9}\]

Step 3: Write the equation of the line through (1, -1) parallel to PS

The slope of the line parallel to \(PS\) will be the same as the slope of \(PS\), which is \(-\frac{2}{9}\). Using point-slope form of the equation of a line, \(y - y_1 = m(x - x_1)\) with the point \((1, -1)\) and slope \(-\frac{2}{9}\): \[y - (-1) = -\frac{2}{9}(x - 1)\]Simplifying, we get:\[9(y + 1) = -2(x - 1)\]Expanding:\[9y + 9 = -2x + 2\]Bringing terms together:\[2x + 9y + 7 = 0\]

Step 4: Choose the correct option

Looking at the options given:- (A) \(4x - 7y - 11 = 0\)- (B) \(2x + 9y + 7 = 0\)- (C) \(4x + 7y + 3 = 0\)- (D) \(2x - 9y - 11 = 0\)The equation we obtained matches option (B): \(2x + 9y + 7 = 0\).

Key Concepts

Median of a Triangle

In coordinate geometry, a median of a triangle is a line segment joining a vertex to the midpoint of the opposite side. To find the median, certain calculations are required that involve understanding the geometric properties of the triangle.
To calculate the median from a vertex, for instance from point \( P \) to the midpoint of \( QR \), follow these steps:

• Identify the coordinates of the endpoints of the side opposite the vertex (\( Q \) and \( R \)).
• Calculate the midpoint of the side using the midpoint formula: \[ \text{Midpoint} = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \].
• Draw a line segment from the vertex to this midpoint, which is the median.

In our scenario, with vertices \( Q(6, -1) \) and \( R(7, 3) \), calculating the midpoint results in \( \left( \frac{13}{2}, 1 \right) \). This midpoint serves as the other endpoint of the median originating from vertex \( P(2, 2) \). The role of calculating a median is crucial in further computations such as finding slopes or equations of lines based on these geometric constructs.
The median serves as a foundational concept in coordinate geometry that helps in graphically representing triangle properties and conducting further analyses.

Slope of a Line

The slope of a line is an important concept in coordinate geometry, representing the tilt or inclination of the line. It's calculated as the rate of change of the y-coordinates with respect to x-coordinates, essentially determining how much the line rises or falls as it moves horizontally. Here's how you calculate it:

• Identify two points on the line. Let's say these points are \( A(x_1, y_1) \) and \( B(x_2, y_2) \).
• Use the slope formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \].

This formula will give you a numerical value indicating the direction and steepness of the line. A positive slope means the line rises as it moves to the right, while a negative slope indicates it falls.
In the original problem, to find the slope of median \( PS \), we used point \( P(2, 2) \) and the midpoint of \( QR \), resulting in \(-\frac{2}{9}\). This indicates a gentle downward slope when moving along the line from left to right.
Knowing the slope is pivotal when creating equations of lines, especially when lines are parallel or perpendicular to each other, as these relationships depend on their slope values.

Equation of a Line

Understanding how to derive the equation of a line is essential in coordinate geometry, especially when dealing with points and slopes. The line's equation provides a mathematical representation of all its points. Here’s a concise explanation:

• The most common form to write the equation of a line is the slope-intercept form: \( y = mx + c \), where \( m \) is the slope and \( c \) is the y-intercept.
• Another useful form is the point-slope form: \( y - y_1 = m(x - x_1) \), ideal when you have a slope and a specific point \( (x_1, y_1) \) on the line.

To derive the equation, take these steps:
- Plug the known slope into the point-slope formula.
- Substitute the coordinates of the known point.
- Simplify to get the equation in a desired form.
In the exercise, we used the point \( (1, -1) \) and slope \(-\frac{2}{9}\) to form the equation: \( 9(y + 1) = -2(x - 1) \). After simplification, this results in \( 2x + 9y + 7 = 0 \), showing how a thorough understanding of point-slope form can easily lead to a line’s equation.
This knowledge empowers you to handle geometrical problems effectively by forming and interpreting various line equations.