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In the xy-coordinate system, rectangle ABCD is inscribed

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In the xy-coordinate system, rectangle ABCD is inscribed

by BTGmoderatorLU » Mon Nov 26, 2018 2:40 pm

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Source: GMAT Prep

In the xy-coordinate system, rectangle ABCD is inscribed within a circle having the equation x^2 + y^2 = 25. Line segment AC is a diagonal of the rectangle and lies on the x-axis. Vertex B lies in quadrant II and vertex D lies in quadrant IV. If side BC lies on line y=3x+15, what is the area of rectangle ABCD?

A. 15
B. 30
C. 40
D. 45
E. 50

The OA is B
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by Jay@ManhattanReview » Mon Nov 26, 2018 11:09 pm
BTGmoderatorLU wrote:Source: GMAT Prep

In the xy-coordinate system, rectangle ABCD is inscribed within a circle having the equation x^2 + y^2 = 25. Line segment AC is a diagonal of the rectangle and lies on the x-axis. Vertex B lies in quadrant II and vertex D lies in quadrant IV. If side BC lies on line y=3x+15, what is the area of rectangle ABCD?

A. 15
B. 30
C. 40
D. 45
E. 50

The OA is B
Pl. see the image which is self-explanatory.

Image

Given that the equation of the circle is x^2 + y^2 = 25, we have the radius of the circle = √25 = 5 and the coordinate of the center of the circle = (0, 0). Thus, AC = diameter = 10.

Area of the rectangle ABCD = 2*Area of the triangle ABC = 2*(1/2*BB'*AC) = BB'*10 = 10y.

We have to get the value of y coordinate of point B.

Since point B is at the circle as well as on the line BC (y = 3x + 15), the coordinate of point B would satisfy both.

Plugging-in y = 3x + 15 in x^2 + y^2 = 25, we get

x^2 + (3x + 15)^2 = 25
x^2 + 9x^2 + 90x + 225 = 25
10x^2 + 90x + 200 = 0
x^2 + 9x + 20 = 0
x^2 + 5x + 4x + 20 = 0

x(x + 5) + 4(x + 5) = 0

(x + 5)(x + 4) = 0
x = - 5 or -4

=> The corresponding values of y are 0 and 3.

Thus, there are two points that lie on the line BC and the circle and their coordinates are (-5, 0) and (-4, 3).
Since it is given that point B lies on the II quadrant, the coordinates of point B would be (-4, 3). This given y = 3

Area of the rectangle ABCD = 10y = 10*3 = 30.

The correct answer: B

Hope this helps!

-Jay
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