One of the diameters of the circle circumscribing the rectangle ABCD is 4y = x + 7. If A and B are t

Q: Why can't we just find the radius of the circle to solve this?

You could calculate the radius, but it is unnecessary work! In coordinate geometry, the center is the heart of the circle. Once you have the center, symmetry allows you to find the other vertices of the rectangle instantly without ever needing the radius.

Q: Why is the perpendicular bisector of $AB$ just $x = 1$?

Great question. Points $A(-3, 4)$ and $B(5, 4)$ share the same $y$-coordinate, meaning the line $AB$ is horizontal. A line perpendicular to a horizontal line must be vertical. Since it passes through the midpoint $M(1, 4)$, the equation is simply $x = 1$.

Q: How do I know the center of the circle is the center of the rectangle?

In a rectangle, the diagonals bisect each other at the rectangle's center. In a circle, the center is the midpoint of any diameter. Since the diagonals of an inscribed rectangle are diameters of the circle, their intersection point must be the circle's center.

Question

One of the diameters of the circle circumscribing the rectangle $ABCD$ is $4y = x + 7$. If $A$ and $B$ are the points $(-3, 4)$ and $(5, 4)$ respectively, then find the area of rectangle.

Sigma Percentile · Accepted Answer

### Analyzing the Setup

Welcome, future engineer! Today, we are going to peel back the layers of a beautiful coordinate geometry problem. It is not just about finding an area; it is about understanding the soul of a rectangle inscribed in a circle.

When you look at a problem like this, do not just see numbers; see the geometric constraints that define the shape.

### The Horizontal Chord

We begin with points $A(-3, 4)$ and $B(5, 4)$. Notice the $y$-coordinates are identical. This is a gift!

It tells us that the chord $AB$ is perfectly horizontal. In the world of coordinate geometry, horizontal and vertical lines are our best friends because they simplify our calculations immensely.

The midpoint $M$ of this chord is simply the average of the coordinates:

$$M = \left( \frac{-3 + 5}{2}, \frac{4 + 4}{2} ight) = (1, 4)$$

### The Perpendicular Bisector

Here is the core geometric reality: the perpendicular bisector of any chord in a circle must pass through the center. Since $AB$ is horizontal, its perpendicular bisector is a vertical line passing through $M(1, 4)$.

Thus, the equation of this line is $x = 1$. This line is the locus of all points equidistant from $A$ and $B$, and it is guaranteed to house the center of our circle.

This is the first pillar of our solution.

### The Intersection

We are given the equation of a diameter: $4y = x + 7$. The center of the circle, let's call it $O$, must lie on this diameter.

But we also know $O$ lies on our perpendicular bisector $x = 1$. This is the intersection point!

By substituting $x = 1$ into the diameter equation, we get:

$$4y = 1 + 7$$

$$4y = 8 \Rightarrow y = 2$$

Our center is $O(1, 2)$. We have successfully located the heart of the circle.

### The Rectangle's Symmetry

In a rectangle inscribed in a circle, the center of the circle is the midpoint of the diagonals. This is the key to finding the remaining vertices $C$ and $D$.

Using the midpoint formula in reverse, we find $C(5, 0)$ and $D(-3, 0)$. It is a beautiful moment of symmetry—the rectangle is perfectly balanced around the center $O(1, 2)$.

### Final Calculation

With the vertices identified, the length of the rectangle is $8$ and the width is $4$.

The area is calculated as:

$$	ext{Area} = 8 	imes 4 = 32$$

We have conquered the problem! Remember, in JEE Advanced, the path is often hidden in the symmetry of the figures. Keep visualizing, keep calculating, and keep falling in love with the math behind the problem. The final answer is **32**.

Animated Solution for Mathematics - Circles: One of the diameters of the circle circumscribing the rectangle $A B C D$ is $4 y = x + 7$ . If $A$ and $B$ are the points $(- 3, 4)$ and $(5, 4)$ respectively, then find the area of rectangle.

Enter Numerical Value:

Visualized Solution

Plotting Given Points $A$ and $B$

Midpoint of $A B$

Perpendicular Bisector of $A B$

The Given Diameter

Finding Center $O$ : Substitution

Finding Center $O$ : Execution

Circumscribing Circle

Properties of Rectangle

Finding Vertices $C$ and $D$

Dimensions of Rectangle

Final Area Calculation

The Sigma Insight: Standard and General Equation of a Circle

Analyzing the Setup

The Horizontal Chord

The Perpendicular Bisector

The Intersection

The Rectangle's Symmetry

Final Calculation

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