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16 Sep 2014, 01:11
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B
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Difficulty:
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Question Stats:
76% (02:07) correct
24%
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If a circle passes through points \((1, 2)\), \((2, 5)\), and \((5, 4)\), what is the diameter of the circle?
A. \(3\sqrt{2}\)
B. \(2\sqrt{5}\)
C. \(\sqrt{22}\)
D. \(\sqrt{26}\)
E. \(\sqrt{30}\)
A. \(3\sqrt{2}\)
B. \(2\sqrt{5}\)
C. \(\sqrt{22}\)
D. \(\sqrt{26}\)
E. \(\sqrt{30}\)
16 Sep 2014, 01:11
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Official Solution:
If a circle passes through points \((1, 2)\), \((2, 5)\), and \((5, 4)\), what is the diameter of the circle?
A. \(3\sqrt{2}\)
B. \(2\sqrt{5}\)
C. \(\sqrt{22}\)
D. \(\sqrt{26}\)
E. \(\sqrt{30}\)
To find the diameter of the circle passing through the given points, we can first find the center of the circle and then find the distance between the center and any one of the given points, which would be equal to the radius of the circle. Finally, the diameter of the circle would be twice the radius.
In an xy-plane, the circle with center \((a, b)\) and radius \(r\) is the set of all points \((x, y)\) such that \((x - a)^2 + (y - b)^2 = r^2\)
Since the given three points are on the circle, they must satisfy the equation of the circle given above. Substituting the coordinates of the given points into the equation of the circle, we get:
\((1 - a)^2 + (2 - b)^2 = r^2\)
\((2 - a)^2 + (5 - b)^2 = r^2\)
\((5 - a)^2 + (4 - b)^2 = r^2\)
Expanding these equations:
\(a^2 - 2a + b^2 - 4b + 5 = r^2\)
\(a^2 - 4a + b^2 - 10b + 29 = r^2\)
\(a^2 - 10a + b^2 - 8b + 41= r^2\)
Subtracting the second equation from the first, and then the third equation from the second, we get:
\(2a +6b = 24\)
\(6a-2b = 12\)
Solving these two equations, we get:
\(a = 3\) and \(b = 3\).
Substituting these values into any of the three equations above, we get:
\(r^2 =(1 - 3)^2 + (2 - 3)^2 \)
\(r^2 = 5\)
\(r = \sqrt{5}\)
Therefore, the diameter of the circle is:
\(2r = 2*\sqrt{5}\)
Answer: B
If a circle passes through points \((1, 2)\), \((2, 5)\), and \((5, 4)\), what is the diameter of the circle?
A. \(3\sqrt{2}\)
B. \(2\sqrt{5}\)
C. \(\sqrt{22}\)
D. \(\sqrt{26}\)
E. \(\sqrt{30}\)
To find the diameter of the circle passing through the given points, we can first find the center of the circle and then find the distance between the center and any one of the given points, which would be equal to the radius of the circle. Finally, the diameter of the circle would be twice the radius.
In an xy-plane, the circle with center \((a, b)\) and radius \(r\) is the set of all points \((x, y)\) such that \((x - a)^2 + (y - b)^2 = r^2\)
Since the given three points are on the circle, they must satisfy the equation of the circle given above. Substituting the coordinates of the given points into the equation of the circle, we get:
\((1 - a)^2 + (2 - b)^2 = r^2\)
\((2 - a)^2 + (5 - b)^2 = r^2\)
\((5 - a)^2 + (4 - b)^2 = r^2\)
Expanding these equations:
\(a^2 - 2a + b^2 - 4b + 5 = r^2\)
\(a^2 - 4a + b^2 - 10b + 29 = r^2\)
\(a^2 - 10a + b^2 - 8b + 41= r^2\)
Subtracting the second equation from the first, and then the third equation from the second, we get:
\(2a +6b = 24\)
\(6a-2b = 12\)
Solving these two equations, we get:
\(a = 3\) and \(b = 3\).
Substituting these values into any of the three equations above, we get:
\(r^2 =(1 - 3)^2 + (2 - 3)^2 \)
\(r^2 = 5\)
\(r = \sqrt{5}\)
Therefore, the diameter of the circle is:
\(2r = 2*\sqrt{5}\)
Answer: B
03 Jun 2023, 03:10
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We can use an alternate method that will be less time consuming.
First we need to find the slopes of the 3 lines that form a triangle inside the circle.
Let the points be A(1,2), B(2,5), C(5,4). And the triangle will be ABC.
Hence the slopes will be as follows:
AB=3, BC=-1/3
Slope of AB*Slope of BC=-1
This implies that AB and BC are perpendicular to each other and AC is the hypotenuse of the triangle ABC.
We know that the diameter of a circle makes a right angle triangle inside the same circle.
Hence AC is nothing but the diameter of the circle.
Lenght of AC=√ (4-2)^2+(5-1)^2
=√ 20
=2√ 5
Option B is the correct answer.
First we need to find the slopes of the 3 lines that form a triangle inside the circle.
Let the points be A(1,2), B(2,5), C(5,4). And the triangle will be ABC.
Hence the slopes will be as follows:
AB=3, BC=-1/3
Slope of AB*Slope of BC=-1
This implies that AB and BC are perpendicular to each other and AC is the hypotenuse of the triangle ABC.
We know that the diameter of a circle makes a right angle triangle inside the same circle.
Hence AC is nothing but the diameter of the circle.
Lenght of AC=√ (4-2)^2+(5-1)^2
=√ 20
=2√ 5
Option B is the correct answer.
