Problem 55 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

Two sides of a rhombus \(A B C D\) are parallel to the lines \(y=x+2\) and \(y=7 x+3\). If the diagonals of the rhombus intersect at the point \((1,2)\) and the vertex \(A\) is on the \(y\)-axis, then the possible coordinates of \(A\) are (A) \((0,0)\) (B) \(\left(0, \frac{5}{2}\right)\) (C) \(\left(0,-\frac{5}{2}\right)\) (D) none of these

Step-by-Step Solution

Answer

(B) \((0, \frac{5}{2})\) and (C) \((0, -\frac{5}{2})\) are possible coordinates of A.

Step 1: Understand the Properties of a Rhombus

A rhombus has all sides of equal length and its diagonals bisect each other at right angles. The diagonals also intersect at the rhombus's center. Here, they intersect at the point \((1, 2)\).

Step 2: Identifying Parallel Sides

The sides of the rhombus are parallel to the lines \(y = x + 2\) and \(y = 7x + 3\). Sides of a rhombus parallel to these lines would have the same slopes, \(m = 1\) and \(m = 7\), respectively.

Step 3: Determine Equations of Diagonals

Since the diagonals bisect each other at point \((1, 2)\) and the rhombus's vertex \(A\) is on the \(y\)-axis, we find that the equations of the diagonals are perpendicular to each set of parallel sides.

Step 4: Setting a Slope Condition for Diagonal

The equation of one of the diagonals can be assumed as \(x = 1\), perpendicular to \(y = x+2)\) and passing through \((1, 2)\).

Step 5: Calculate the Point of Intersection of the Diagonal

Since the diagonal \(x = 1\) is vertical, the second diagonal will be horizontal to satisfy the right-angle condition, intersecting the \(y\)-axis at \(y = 2\). Hence, \(A = (0, y)\).

Step 6: Determine the Length of Half the Diagonal

Given that \((1, 2)\) is the center, check the vertical distance between \((1, 2)\) and \((0, y)\), which must be symmetric passing through this center: \(|2 - y| = |y - 2|\). Solving for this gives \(y = \frac{5}{2}\) or \(-\frac{5}{2}\).

Step 7: Check Given Options

Aligning our values of \(y\) with the options provided, it matches option (B) \((0, \frac{5}{2})\) and option (C) \((0, -\frac{5}{2})\). Therefore, these could possibly be correct answers.

Key Concepts

Properties of Rhombus

A rhombus is a special type of quadrilateral with unique features. Each side of a rhombus is of equal length, giving it symmetrical properties. One of the most distinctive characteristics of a rhombus is that its diagonals intersect each other at right angles, dividing the rhombus into four right-angled triangles. This bisection means that the diagonals split each other into two equal halves right at the center of the rhombus. This central intersection point is not only the midpoint of the diagonals but also serves as a crucial point for verifying coordinates or other properties within problems involving rhombuses.
Understanding these basic properties helps visualize how elements like diagonals and sides interact within the rhombus, enabling easier problem-solving.

Parallel Lines

In coordinate geometry, parallel lines have the same slope. This concept is applied in problems involving rhombuses where the sides are parallel to given lines. In this specific exercise, two sides of the rhombus are parallel to the lines given by the equations \(y = x + 2\) and \(y = 7x + 3\).
This implies that the parallel sides of the rhombus inherit the slopes of these lines, meaning one pair of opposite sides will have a slope of 1, and another pair will have a slope of 7. Understanding the slopes involved is critical because it guides us in formulating the equations of the sides and diagonals within the realm of coordinate geometry.

Diagonals of Rhombus

The diagonals of a rhombus carry unique properties key to solving geometric problems. Firstly, these diagonals always bisect each other at right angles. In the exercise, the diagonals intersect at the point \((1, 2)\).
To find the equations of these diagonals, we use the intersection point as a reference. The problem reveals that one diagonal is vertical, suggesting that one equation is \(x = 1\), with the vertical direction ensuring perpendicularity to one set of parallel sides. The opposition diagonal, being horizontal to fulfill the perpendicularity criterion, intersects the \(y\)-axis at \(y = 2\).
These insights reflect the importance of understanding the bisecting nature and perpendicular intersection of the rhombus's diagonals.

Intersection Points

In this problem, understanding intersection points is crucial. The given intersection point of the diagonals is \((1, 2)\). This point divides each diagonal into two equal lengths and serves as an anchor for calculating vertex positions.
For instance, it helps us determine that a vertex lies on the \(y\)-axis by using symmetry. Given that the intersection point is the midpoint, the vertical distance to any vertex on the \(y\)-axis is symmetric relative to this midpoint. Solving the equation \(|y - 2| = |2 - y|\) derives possible \(y\)-coordinate values of \( \frac{5}{2}\) and \(-\frac{5}{2}\).
This demonstrates how intersection points can be leveraged to solve for unknowns in a coordinate plane setup.