Problem 63 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

Two sides of a parallelogram are along the lines, \(x+y=3\) and \(x-y+3=0\). If its diagonals intersect at \((2,4)\), then one of its vertex is: \(\quad\) [Jan. 10, 2019 (II)] (a) \((3,5)\) (b) \((2,1)\) (c) \((2,6)\) (d) \((3,6)\)

Step-by-Step Solution

Answer

One vertex of the parallelogram is (2,1).

Step 1: Identify the Equation of Lines

The sides of a parallelogram are on the lines given by the equations: 1. Line 1: \(x + y = 3\) 2. Line 2: \(x - y + 3 = 0\) or equivalently \(x - y = -3\).

Step 2: Find the Intersection Points

To find the coordinates of the vertices of the parallelogram, we need to solve these equations. Solve the system:1. Solve for \(x\) in terms of \(y\) from Line 1: \[ x = 3-y \]2. Substituting in Line 2: \[ (3-y) - y = -3 \] \[ 3 - 2y = -3 \] \[ -2y = -6 \] \[ y = 3 \]3. Substitute back to find \(x\): \[ x = 3-3 = 0 \]So, the intersection point is \((0,3)\).

Step 3: Apply the Diagonal Intersection Information

Since the diagonals of a parallelogram bisect each other, if they intersect at point \((2,4)\), then this point is the midpoint of the diagonal between the vertex we want and its opposite vertex. For simplicity, assume one vertex is \((a, b)\), then the opposite vertex is \((4-a, 8-b)\) by the midpoint formula. However, to find the exact vertices, substitute potential options into the lines.

Step 4: Substitute Potential Vertex Points into Line Equations

We need to check which of the provided options is a valid vertex. We substitute the potential answers into the line equations:1. Option (a) \((3,5)\): - Check for Line 1: \(x+y=3\) \[ 3+5 = 8, \text{ not on line 1}\] - Check for Line 2: \(x-y+3=0\) \[ 3-5+3 = 1, \text{ not on line 2}\]Continuing similarly for each point...2. Option (c) \((2,6)\): - Check for Line 1: \(x+y=3\) \[ 2+6 = 8, \text{ not on line 1}\] - Check for Line 2: \(x-y+3=0\) \[ 2-6+3 = -1, \text{ not on line 2}\]3. Option (d) \((3,6)\): - Check for Line 1: \(x+y=3\) \[ 3+6 = 9, \text{ not on line 1}\] - Check for Line 2: \(x-y+3=0\) \[ 3-6+3 = 0, \text{ lies on line 2}\]4. Option (b) \((2,1)\): - Check for Line 1: \(x+y=3\) \[ 2+1 = 3, \text{ lies on line 1}\] Option (b) \((2,1)\) satisfies one of the lines.

Key Concepts

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is a branch of mathematics that uses algebraic equations to describe geometric shapes and their properties in a coordinate plane.
This approach allows us to study shapes such as lines, circles, polygons, and more complex figures using numeric coordinates.

• Each point in the plane is described by an ordered pair of numbers known as coordinates \((x, y)\).
• The coordinates describe the position of a point in relation to two intersecting lines known as axes, which typically form a right angle.
• The x-axis runs horizontally, while the y-axis runs vertically.

In the case of a parallelogram, the coordinates of its vertices can be found by solving equations of lines that form its sides.
By understanding the relation between lines and points in the plane, coordinate geometry enables us to explore many properties of geometric figures.

Diagonal Intersection

The intersection of diagonals in a parallelogram is a key property that simplifies many calculations.
In every parallelogram, the diagonals bisect each other, meaning the point where they intersect is the midpoint of both diagonals.

• This implies that if a parallelogram has a diagonal intersection at \((2, 4)\), then each diagonal divides the other into two equal parts at this point.
• This can be used to find missing vertices: if one vertex is assumed as \((a, b)\), then its opposite vertex can be mathematically represented by \((4-a, 8-b)\).

Using the midpoint formula helps in determining the unknown vertices when provided with certain known points, making diagonal intersection a powerful tool in coordinate geometry.

Parallelogram Properties

Parallelograms are fundamental shapes with unique properties that distinguish them from other quadrilaterals.
Understanding these properties is essential in solving geometric problems involving parallelograms.

• The opposite sides of a parallelogram are equal in length and parallel.
• The opposite angles are equal in size.
• The diagonals bisect each other, as discussed, and this property is used significantly in coordinate geometry problems.

In the context of coordinate geometry, the properties can be validated by checking relationships between line equations:
ensuring that pairs of lines are parallel or checking that the diagonals bisect at a given midpoint aids in confirming that a quadrilateral is a parallelogram.

Line Equations

Line equations are at the heart of coordinate geometry as they describe the infinite set of points that form straight lines on the coordinate plane.
These equations can be written in various forms, with the slope-intercept form \(y = mx + c\) being one of the most popular.

• In this exercise, the sides of the parallelogram are given by the line equations \(x + y = 3\) and \(x - y = -3\).
• These represent lines in a coordinate plane and can be rearranged for different problem-solving scenarios.
• Solving such systems of linear equations can reveal the points of interception—potential vertices of geometric shapes.

For instance, solving for \(x\) and \(y\) can determine the coordinates of intersection points, helping to visualize and locate vertices of shapes like parallelograms on the coordinate grid.