Problem 60 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

A rectangle is inscribed in a circle with a diameter lying along the line \(3 y=x+7\). If the two adjacent vertices of the rectangle are \((-8,5)\) and \((6,5)\), then the area of the rectangle (in sq. units) is: [April 09, 2019 (II)] (a) 84 (b) 98 (c) 72 (d) 56

Step-by-Step Solution

Answer

The area of the rectangle is 98 square units.

Step 1: Identify the orientation of the rectangle

Since the diameter of the circle is along the line \( 3y = x + 7 \), we note this is a straight line and provides the direction in which the center of the circle is positioned. The rectangle inscribed in the circle means the diagonal of the rectangle is equal to the diameter of the circle.

Step 2: Find the midpoint of given vertices

The two given vertices of the rectangle are \((-8,5)\) and \((6,5)\). The midpoint can be calculated as \( \left( \frac{-8+6}{2}, \frac{5+5}{2} \right) = (-1, 5) \).

Step 3: Determine the length of vertices on x-axis

Since the two points share the same y-coordinate \(5\), they lie horizontally; thus the length \(a\) of this side of the rectangle is the difference in the x-coordinates: \( |6 - (-8)| = 14 \).

Step 4: Use properties of a rectangle inscribed in a circle

For a rectangle inscribed in a circle, the diameter of the circle is the diagonal of the rectangle. With one side known, use the Pythagorean theorem for rectangles: if the lengths of the sides are \( a \) and \( b \), the diagonal is given by \( \sqrt{a^2 + b^2} \). Here, diagonal \(= \sqrt{14^2 + b^2} \).

Step 5: Set and solve equation for diameter

The diameter of the circle can also be calculated from the line. Using point \((-8,5)\), the equation of the circle centered at \((-1,5)\) with a known diagonal \(=14\) gives the circle's radius \(= \frac{14}{2} = 7\). Thus, \( b = \sqrt{7^2 - 14^2 + b^2} = \sqrt{49 - 196} = 7\).

Step 6: Compute the area of the rectangle

The area \(A\) of the rectangle is given by \( a \times b \). Here both sides are \(14\) and \(b=7\), so area \(A = 14 \times 7 = 98\).

Key Concepts

Rectangles

Rectangles are four-sided shapes or quadrilaterals, where opposite sides are parallel and equal in length. They have four right angles, each measuring 90 degrees.
This symmetry makes rectangles a fundamental shape in geometry. Understanding their properties is essential for solving geometry-related problems.

• Opposite sides are equal and parallel.
• All internal angles are 90 degrees.
• The diagonals of a rectangle are equal in length.

In problems involving rectangles—like the one discussed—it's crucial to identify the sides and diagonal. In this particular problem, the two given points have the same y-coordinate, indicating the line is horizontal. Hence, the horizontal side length is simply the difference between the x-coordinates of these points (14 units here). Recognizing these properties helps in calculating the area and solves geometry problems efficiently.

Circles

Circles are round shapes that are equally distant from a central point called the center. The distance from this center to any point on the circle is the radius, and twice the radius is the diameter.
In coordinate geometry, a circle inscribed around a shape, like a rectangle, implies that the shape's diagonal equals the circle's diameter.

• The circle's center aligns with the midpoint of the diagonal of an inscribed rectangle.
• Properties of the circle, like its radius and diameter, are fundamental in problem-solving.
• An inscribed rectangle's diagonal acts as the circle's diameter.

In our given problem, the circle's diameter is identified by using the line equation and the rectangle's properties. This relationship helps us find the lengths critical to solving for the rectangle's area in such exercises.

Pythagorean Theorem

The Pythagorean Theorem is a vital formula in geometry, especially concerning right triangles. It states that in a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
This theorem is especially helpful when dealing with rectangles, as their diagonals can be treated like hypotenuses.
The formula is given as:\[ c = \sqrt{a^2 + b^2}\]Where:

• \( c \) is the hypotenuse.
• \( a \) and \( b \) are the other two sides.

This formula is applied in the problem to find the diagonal of the rectangle by using the lengths of its sides. Solving for one side using the diagonal calculated from the circle's diameter showcases its practical application in geometry problems.

Geometry Problems

Geometry problems often involve finding unknown values, such as lengths, areas, or angles, using known properties and theorems. Approaching these problems requires logical thinking and a systematic method to ensure all geometric relationships and properties are considered.
To solve complex problems like the one described, where a rectangle is inscribed in a circle, you need a strong grasp of related concepts.
Here are some steps to approach such problems:

• Identify given information and known relationships between geometric figures.
• Utilize geometric properties, such as symmetry or parallel lines, to simplify calculations.
• Apply relevant theorems, like the Pythagorean theorem, to find unknown values.
• Ensure calculations reflect the real geometric scenario described in the problem.

By interconnecting concepts such as the properties of rectangles and circles, and applying them effectively through problem-solving strategies in coordinate geometry, students can enhance their skills and confidently approach similar geometry problems.