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in last wy are we myltiplyin by 1/2
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in last wy are we myltiplyin by 1/2
Question is asking us for area in first quadrant only. Since area enclosed by in first quandrant = area enclosed in second quadrant we are dividing the total area by 2, hope this is clear
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In the coordinate plane, a circle with its center on the negative x-axis has a radius of 12 units, and passes through (0, 6) and (0, – 6). What is the area of the part of this circle in the first quadrant?
(A) \(6\pi - 12\sqrt{3}\)
(B) \(12\pi - 12\sqrt{3}\)
(C) \(12\pi - 18\sqrt{3}\)
(D) \(24\pi - 18\sqrt{3}\)
(E) \(24\pi - 36\sqrt{3}\)


This is a question from a set of 15 challenging GMAT Quant practice questions. For the whole collection, as well as the OE with diagram for this question, see:
Challenging GMAT Math Practice Questions

Mike :-)

We need the area OPMO.
People with some trigonometry knowledge can solve this fairly quickly by recognising that Sin of angle PCO is 1/2 which means that angle PCO is 30 degrees.
Attachment:
Screenshot 2023-03-25 at 12.45.48 PM.png
Screenshot 2023-03-25 at 12.45.48 PM.png [ 38.95 KiB | Viewed 2100 times ]
Area OPMO = Area of sector CPMC - Area of triangle CPO = \(\frac{30}{360}* \pi * 12^2 - \frac{1}{2}*6*\sqrt{12^2-6^2}\)
Area OPMO = \(12\pi - 18\sqrt{3}\)

Answer (C)

In case one doesn't want to use trigonometry, one could recognize that triangle CPQ is an equilateral triangle. So angle PCQ is 60 degrees which means angle PCO is 30 degrees and then proceed as before.
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