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

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

Step 1: Find the midpoint of QR

The median from vertex \(P\) will go through the midpoint of \(QR\). To find the midpoint \(M\) of \(QR\), use the midpoint formula: \(M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)\). For points \(Q(6, -1)\) and \(R(7, 3)\), the midpoint is \(M = \left( \frac{6 + 7}{2}, \frac{-1 + 3}{2} \right) = (6.5, 1)\).

Step 2: Determine the slope of PS

The median \(PS\) connects vertex \(P(2,2)\) and the midpoint \(M(6.5, 1)\). The slope \(m\) of a line through two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by \(m = \frac{y_2 - y_1}{x_2 - x_1}\). Thus, the slope of \(PS\) is \(m = \frac{1 - 2}{6.5 - 2} = \frac{-1}{4.5} = -\frac{2}{9}\).

Step 3: Find the equation of line parallel to PS

A line parallel to \(PS\) will have the same slope \(m = -\frac{2}{9}\). The line passes through the point \((1, -1)\). Use the point-slope form \(y - y_1 = m(x - x_1)\) to find the equation. Substituting the values, we get \(y + 1 = -\frac{2}{9}(x - 1)\). Simplifying, \(9(y + 1) = -2(x - 1) \Rightarrow 9y + 9 = -2x + 2 \Rightarrow 2x + 9y = -7\).

Step 4: Compare with given options

Rewrite the final equation \(2x + 9y + 7 = 0\) and compare it with the options provided:(A) \(4x - 7y - 11 = 0\)(B) \(2x + 9y + 7 = 0\)(C) \(4x + 7y + 3 = 0\)(D) \(2x - 9y - 11 = 0\)The equation matches option (B), \(2x + 9y + 7 = 0\).

Key Concepts

Midpoint Formula

The midpoint formula is a useful tool in coordinate geometry. It helps us find the exact center point between two given points on a coordinate plane. This concept is essential when trying to determine points such as medians in triangles.

To calculate the midpoint between two points, let's say \( (x_1, y_1) \) and \( (x_2, y_2) \), you can use the formula:

• \( M = \left ( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right ) \)

This formula works by averaging the x-coordinates to find the x of the midpoint, and averaging the y-coordinates to find the y of the midpoint.

For example, if we have the two points \( Q(6, -1) \) and \( R(7, 3) \), we can use the formula to find the midpoint. Simply plug in the values:

• \( M = \left ( \frac{6 + 7}{2}, \frac{-1 + 3}{2} \right ) = (6.5, 1) \)

Thus, the midpoint \( M \) is \( (6.5, 1) \). This simple calculation tells us that between points \( Q \) and \( R \), this is the center point.

Slope of a Line

Understanding the slope of a line is crucial in coordinate geometry as it shows us how steep the line is and the direction it goes.

The slope \( m \) can be found for any line passing through two points \( (x_1, y_1) \) and \( (x_2, y_2) \) with the formula:

• \( m = \frac{y_2 - y_1}{x_2 - x_1} \)

This formula takes the change in y (vertical change) and divides it by the change in x (horizontal change).

For example, consider the points \( P(2,2) \) and \( M(6.5,1) \). To find the slope of the line \( PS \), apply the formula:

• \( m = \frac{1 - 2}{6.5 - 2} = \frac{-1}{4.5} = -\frac{2}{9} \)

Therefore, the slope of the line connecting \( P \) and \( M \) is \(- \frac{2}{9} \). This negative value indicates the line descends as you move from left to right.

Equation of a Line

The equation of a line in coordinate geometry allows us to represent a line algebraically and understand its properties and path.

A commonly used form to find the equation of the line is the point-slope form, which is:

• \( y - y_1 = m(x - x_1) \)

Here, \( m \) is the slope, and \( (x_1, y_1) \) is a specific point the line passes through.

For example, if we know a line is parallel to \( PS \) with a slope of \( -\frac{2}{9} \), and it passes through point \( (1, -1) \), we use:

• \( y + 1 = -\frac{2}{9}(x - 1) \)

To simplify, multiply through by 9 to clear the fraction:

• \( 9(y + 1) = -2(x - 1) \)
• \( 9y + 9 = -2x + 2 \)
• \( 2x + 9y = -7 \)

The final line equation when rewritten is \( 2x + 9y + 7 = 0 \). This shows us the exact path and position of the line on the coordinate plane.