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