If the length of the diameter of a circle is equal to the...

If the length of the diameter of a circle is equal to the length of the major axis of the ellipse whose equation is $x^{2}+4 y^{2}-4 x+8 y-28=0,$ to the nearest whole number, what is the area of the circle?
(A) 28
(B) 64
(C) 113
(D) 254
(E) 452

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Key Concepts

Ellipse

An ellipse is a type of conic section defined by an equation that typically takes the form (x-h)²/a² + (y-k)²/b² = 1, where (h, k) is the center, and a and b represent the lengths of the semi-axes. Understanding this form allows one to extract geometric properties such as the lengths of the major and minor axes.

Completing the Square

Completing the square is an algebraic method used to rewrite quadratic expressions in a way that reveals the structure of a conic section's equation. This technique is crucial for transforming general quadratic equations into their standard forms, thereby making it easier to identify key features like the center and axes lengths of an ellipse or circle.

Major Axis

The major axis of an ellipse is the longest segment that passes through the center, defining the ellipse's maximum width. It is directly related to the semi-major axis, which is half the length of the major axis; knowing this allows one to compare dimensions of different shapes, such as linking the ellipse to a circle.

Circle Area Calculation

The area of a circle is determined by the formula A = ?r², where r is the radius. Accurately calculating the area involves first deducing the radius from given information, such as the diameter, and then applying this formula to compute the final area.

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