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29 Oct 2025, 06:23
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D
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
55%
(hard)
Question Stats:
58% (01:42) correct
42%
(02:01)
wrong
based on 33
sessions
History
Date
Time
Result
Not Attempted Yet
In the x-y plane, find the area (in sq. units) of the region enclosed by the graph of: \(\lvert x-2 \rvert + \Bigl\lvert \frac{1}{2}(y+4) \Bigr\rvert = 4\)
A) 8
B) 16
C) 32
D) 64
E) 128
A) 8
B) 16
C) 32
D) 64
E) 128
09 Nov 2025, 04:28
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Use this formula
|aX-n|+|bY-n|=r
Area= 2r^2/|ab|
Plug in the values and you get 2 x 16 x 2
Hence 64 -> Ans is D
|aX-n|+|bY-n|=r
Area= 2r^2/|ab|
Plug in the values and you get 2 x 16 x 2
Hence 64 -> Ans is D
10 Nov 2025, 01:48
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A general equation of the form a|x − h| + b|y − k| = c represents a rhombus in the coordinate plane.
The constants h and k decide the position of the center, while a and b affect how stretched or compressed the shape is horizontally and vertically.
In this equation, the expressions (x − 2) and (y + 4) indicate that the graph is centered at (2, −4).
Multiplying through by 2 gives 2|x − 2| + |y + 4| = 8, which makes it easier to see how the x- and y-terms scale.
To locate the endpoints of the diagonals, we substitute values that make one absolute term zero:
When x = 2, the first term becomes zero, so |y + 4| = 8.
This gives y = 4 and y = −12, which are the top and bottom vertices.
The vertical diagonal connects (2, 4) and (2, −12), so its length is 16.
When y = −4, the second term becomes zero, so 2|x − 2| = 8.
This gives x = 6 and x = −2, which are the right and left vertices.
The horizontal diagonal connects (6, −4) and (−2, −4), so its length is 8.
Because the figure is symmetric about both axes through its center, it forms a rhombus.
The area of a rhombus equals one-half the product of its diagonals.
Area = (1/2) × 16 × 8 = 64.
The region enclosed by the graph has an area of 64 square units.
Hence, D is the correct answer choice.
The constants h and k decide the position of the center, while a and b affect how stretched or compressed the shape is horizontally and vertically.
In this equation, the expressions (x − 2) and (y + 4) indicate that the graph is centered at (2, −4).
Multiplying through by 2 gives 2|x − 2| + |y + 4| = 8, which makes it easier to see how the x- and y-terms scale.
To locate the endpoints of the diagonals, we substitute values that make one absolute term zero:
When x = 2, the first term becomes zero, so |y + 4| = 8.
This gives y = 4 and y = −12, which are the top and bottom vertices.
The vertical diagonal connects (2, 4) and (2, −12), so its length is 16.
When y = −4, the second term becomes zero, so 2|x − 2| = 8.
This gives x = 6 and x = −2, which are the right and left vertices.
The horizontal diagonal connects (6, −4) and (−2, −4), so its length is 8.
Because the figure is symmetric about both axes through its center, it forms a rhombus.
The area of a rhombus equals one-half the product of its diagonals.
Area = (1/2) × 16 × 8 = 64.
The region enclosed by the graph has an area of 64 square units.
Hence, D is the correct answer choice.
