The line 2 x-y+4=0 is a diameter of the circle which...

The line $2 \mathrm{x}-\mathrm{y}+4=0$ is a diameter of the circle which circumscribes a rectangle $\mathrm{ABCD}$. If the co-ordinates of $\mathrm{A}$ and $B$ are $A(4,6), B(1,9)$, then the area of rectangle $A B C D$ is
(a) 18
(b) 8
(c) 16
(d) 12

Key Concepts

Circumscribed Rectangle

A rectangle is circumscribed by a circle when all four of its vertices lie on the circle. This property implies that the rectangle’s diagonals act as diameters of the circle, and they are equal in length and bisect each other at the circle’s center.

Rectangles and Diagonals

In a rectangle, the diagonals are congruent and intersect at the midpoint, which in the case of a circumscribed rectangle is the center of the circle. This central point is critical in determining other dimensions and characteristics of the rectangle.

Diameter as a Line of the Circle

When a given line is stated to be a diameter of the circle, it directly passes through the circle’s center. This fact provides a key condition: the center must satisfy the equation of the line, which can help in determining the circle’s and, by extension, the rectangle’s dimensions.

Coordinate Geometry Techniques

Employing coordinate geometry, including the use of the distance formula, midpoint formula, and slope calculations, is essential for deducing lengths, verifying perpendicularity (as required in a rectangle), and computing areas from given vertex coordinates.

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