In the xy-coordinate system, rectangle ABCD is inscribed

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.



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
_________________
Manhattan Review GMAT Prep

Locations: Manhattan Review Dilsukhnagar | Hyderabad GMAT Prep | Bangalore GMAT Courses | Kukatpally GRE Prep | and many more...

Schedule your free consultation with an experienced GMAT Prep Advisor! Click here.