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In the coordinate plane, a circle has center (2, -3)

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by BTGmoderatorRO » Thu Mar 01, 2018 12:51 pm
In the coordinate plane, a circle has center (2, -3) and passes through the point (5, 0). What is the area of the circle?

A. 3Ï€
B. 3√2π
C. 3√3π
D. 9Ï€
E. 18Ï€
OA is E
Experts, Can you give me some help here? Please.<i class="em em-raised_hands"></i>
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by Vincen » Sat Mar 03, 2018 3:27 am
Since the area of a circle is $$\pi\cdot r^2$$ we have to find r.

Using the formula of distance between two points in the plane, we have that $$r=d\left(\left(2,-3\right);\left(5,0\right)\right)=\sqrt{\left(5-2\right)^2+\left(0-\left(-3\right)\right)^2}=\sqrt{3^2+3^2}$$ $$\Rightarrow\ r=\sqrt{9\cdot2}=3\sqrt{2}$$ and therefore the area is $$A=\ \pi\cdot\left(3\sqrt{2}\right)^2=18\pi$$ Hence, the answer is E.
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by Scott@TargetTestPrep » Mon Jun 10, 2019 6:21 pm
BTGmoderatorRO wrote:In the coordinate plane, a circle has center (2, -3) and passes through the point (5, 0). What is the area of the circle?

A. 3Ï€
B. 3√2π
C. 3√3π
D. 9Ï€
E. 18Ï€
OA is E
Experts, Can you give me some help here? Please.<i class="em em-raised_hands"></i>
Because the circle passes through the point (5,0) and has center (2,-3), we know that the distance between these points is the radius of the circle.

Since we are given the coordinates for each point, the easiest thing to do is to use the distance formula to determine the circle's radius. The distance formula is:

Distance= √[(x2 - x1)^2 + (y2 - y1)^2]

We are given two ordered pairs, so we can label the following:

x1 = 2
x2 = 5

y1 = -3
y2 = 0

When we plug these values into the distance formula, we have:

Distance= √[(5 - 2)^2 + (0 - (-3))^2]

Distance= √ [(3)^2 + (3)^2]
Distance= √ [9 + 9]

Distance = √ [18]

Distance = √9 x √2

Distance = 3 x √2

Thus, we know that the radius = 3 x √2.

Finally, we can use the radius to determine the area of the circle.

area = πr^2

area = π(3 x √2 )^2

area = π(9 x 2 )

area = 18Ï€

Answer: E

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