Problem 69 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

Let \(P S\) be the median of the triangle vertices \(P(2,2), Q(6,-1)\) and \(R(7,3)\). The equation of the line passing through \((1,-1)\) and parallel to PS is: (a) \(4 x+7 y+3=0\) (b) \(2 x-9 y-11=0\) (c) \(4 x-7 y-11=0\) (d) \(2 x+9 y+7=0\)

Step-by-Step Solution

Answer

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

Step 1: Calculate the coordinates of the midpoint S

The first step is to find the midpoint of the segment \(QR\), which is denoted as \(S\). Using the midpoint formula for points \(Q(6,-1)\) and \(R(7,3)\), calculate \(S\):\[ S\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) = \left( \frac{6 + 7}{2}, \frac{-1 + 3}{2} \right) = \left( \frac{13}{2}, 1 \right). \] So, \(S\) has coordinates \(\left( \frac{13}{2}, 1\right)\).

Step 2: Find the slope of line PS

Now we calculate the slope of the median \(PS\), where \(P(2,2)\) and \(S\left( \frac{13}{2}, 1 \right)\) are endpoints. The slope formula is: \[ m = \frac{y_2-y_1}{x_2-x_1} = \frac{1-2}{\frac{13}{2}-2} = \frac{-1}{\frac{9}{2}} = -\frac{2}{9}. \] The slope of line \(PS\) is \(-\frac{2}{9}\).

Step 3: Determine the equation of the parallel line

A line parallel to \(PS\) has the same slope \(-\frac{2}{9}\) and, in this case, passes through \((1, -1)\). Use the point-slope formula \(y - y_1 = m(x - x_1)\):\[ y + 1 = -\frac{2}{9}(x - 1). \] Multiply through by 9 to clear the fraction: \[ 9(y + 1) = -2(x - 1). \] Simplify to get: \[ 9y + 9 = -2x + 2. \] Rearrange: \[ 2x + 9y = -7. \] So, the equation of the parallel line is \(2x + 9y + 7 = 0\).

Key Concepts

Slope of a Line

The slope of a line provides a measure of its steepness and direction. It's often represented by \( m \). The slope is calculated as the "rise" over the "run"—essentially, how much the line goes up (or down) for each unit it goes right. Mathematically, this is expressed by the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] Where \((x_1, y_1)\) and \((x_2, y_2)\) are coordinates of two distinct points on the line.

• If the slope is positive, the line inclines upwards as you move from left to right.
• If it's negative, the line inclines downwards.

A zero slope indicates a horizontal line, while an undefined slope corresponds to a vertical line. Calculating the slope is fundamental in understanding the equation of a line and its graphical interpretation.

Midpoint Formula

The midpoint formula is a straightforward method for finding the point exactly halfway between two other points on a coordinate plane. Given two endpoints \((x_1, y_1)\) and \((x_2, y_2)\), the coordinates for the midpoint \(M\) are calculated by: \[ M \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \] For example, if you have the points \((6, -1)\) and \((7, 3)\), the midpoint \(M\) will be:

• \( \left( \frac{6 + 7}{2}, \frac{-1 + 3}{2} \right) = \left( \frac{13}{2}, 1 \right) \)

Thus, the midpoint allows us to find the location in the middle of two points, which is particularly useful in various geometric calculations, such as determining medians of triangles.

Parallel Lines

Parallel lines are two lines in a plane that never meet. They remain the same distance apart and have identical slopes. That's the key feature of parallel lines—they have the same slope \( m \). For example, if you have a line with slope \(-\frac{2}{9}\), any line parallel to it will also have a slope of \(-\frac{2}{9}\). When finding the equation of a line parallel to a given line, you'll ensure the slopes match:

• Use the point-slope formula \(y - y_1 = m(x - x_1)\), where \((x_1, y_1)\) is a point on the new line and \(m\) is the given slope.

This concept is particularly useful in problem-solving when you need to establish relationships between lines without them ever intersecting.

Coordinate Geometry

Coordinate geometry, or analytical geometry, blends algebra and geometry to describe geometric figures and solve spatial problems using a coordinate system. It allows for precise calculations and graphical representations of shapes and lines. Basic tools in coordinate geometry include:

• Points: Represented as coordinates \((x, y)\) on a plane.
• Lines: Described by equations, such as \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.
• Distance Formula: Calculates the distance between two points.
• Midpoint Formula: Finds the midpoint between two points.

Through these concepts, coordinate geometry becomes a powerful tool for solving practical problems and visualizing algebraic equations.