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In a rectangle ABCD, the coordinates of A and B are (1,2) and (3,6) respectively and some diameter of the circumscribing circle of ABCD has equation 2x−y+4=0. Then the area of the rectangle is.

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210

25

answer is A.

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Detailed Solution

The correct answer is option 1.
On using the distance formula:

A B = (3 − 1) 2 + (6 − 2) 2 = 22 + 42 = 4 + 16 = 20

We know, that the slope of a line is

− a b

Therefore, the slope of diameter of given equation

2 x − y + 4 = 0

is:

− 2 − 1 = 2

The slope of line joining

A (1, 2)

and

B (3, 6)

6 − 2 3 − 1 = 42 = 2

On drawing the figure:

To find BE using the point to line distance formula,

B E = 2 (3) − 6 + 4 22 + (− 1) 2 = 45

Therefore, length of BC=

2 B E = 2 × 45 = 85

Area of rectangle:

A r e a = A B ⋅ B C = 25 – 85 = 16

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