Circles - GMAT Quantitative Reasoning
Card 0 of 8
What is the equation of a given circle on the coordinate plane?
Statement 1: Its center is at the origin.
Statement 2: One of its diameters has endpoints 
and 
.
What is the equation of a given circle on the coordinate plane?
Statement 1: Its center is at the origin.
Statement 2: One of its diameters has endpoints
You cannot determine the equation of a circle knowing only the center, as in Statement 1.
But given the coordinates of any diameter of a circle, as in Statement 2, you can use the midpoint formula to find the center, and the distance formula to find the distance from the center to either endpoint - this is the radius. Once you know the center and the radius, just apply them to the standard form of the equation of a circle.
You cannot determine the equation of a circle knowing only the center, as in Statement 1.
But given the coordinates of any diameter of a circle, as in Statement 2, you can use the midpoint formula to find the center, and the distance formula to find the distance from the center to either endpoint - this is the radius. Once you know the center and the radius, just apply them to the standard form of the equation of a circle.
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The graph of the equation
,
where 
, is a circle. In which quadrant is the center of the circle located?
Statement 1: 
Statement 2: 
The graph of the equation
where
Statement 1:
Statement 2:
The center of the circle of the equation 
is the point 
.
If Statement 2 is assumed, then 
; since 
is known to be positive, 
is negative, which is the same as Statement 1.
From either statement, we know 
to be negative, and, subsequently, that both coordinates of the center 
are negative, putting it in Quadrant III.
The center of the circle of the equation
If Statement 2 is assumed, then
From either statement, we know
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Find the equation of circle T.
I) Circle T is centered at the point 
.
II) Circle T touches the 
axis at the point 
.
Find the equation of circle T.
I) Circle T is centered at the point
II) Circle T touches the
The equation of a circle is given by:
Where r is the radius and (h,k) are the center of the circle
I) gives us (h,k)
By using II) and distance formula, we can find r
Thus, both statements are needed!
The equation of a circle is given by:
Where r is the radius and (h,k) are the center of the circle
I) gives us (h,k)
By using II) and distance formula, we can find r
Thus, both statements are needed!
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Write the equation for circle 
, a circle in the standard x-y plane.
I) 
is centered at 
.
II) The line 
is tangent to the circle at the point 
.
Write the equation for circle
I)
II) The line
To write the equation of a circle we need the coordinates of its center and the length of its radius.
I) Gives us the center.
II) Tells us that the circle just touches the x-axis at the point (7,0), which is exactly 7 units below our center. This makes our radius 7.
Thus, we need both statements.
To write the equation of a circle we need the coordinates of its center and the length of its radius.
I) Gives us the center.
II) Tells us that the circle just touches the x-axis at the point (7,0), which is exactly 7 units below our center. This makes our radius 7.
Thus, we need both statements.
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Find the equation of Circle X.
I) The x-coordinate of Circle X's center is twice the y-coordinate of Circle X's center.
II) Circle X has a radius of 16 units.
Find the equation of Circle X.
I) The x-coordinate of Circle X's center is twice the y-coordinate of Circle X's center.
II) Circle X has a radius of 16 units.
The equation of a circle is as follows:
Where 
is the center of the circle and 
is the radius.
Statement I tells us how to relate 
and 
, but doesn't tell us how to find either one.
Statement II gives us 
.
We almost have enough information here, but we can't tell for certain where the circle's center is, so more information is needed.
The equation of a circle is as follows:
Where
Statement I tells us how to relate
Statement II gives us
We almost have enough information here, but we can't tell for certain where the circle's center is, so more information is needed.
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Consider Circle L.
I) The radius of Circle L is twice the value of the third prime number when counting the prime numbers and increasing after zero.
II) The x-coordinate of Circle L is four times the y-coordinate of L. The y-coordinate is equal to the radius.
Find the equation of Circle L.
Consider Circle L.
I) The radius of Circle L is twice the value of the third prime number when counting the prime numbers and increasing after zero.
II) The x-coordinate of Circle L is four times the y-coordinate of L. The y-coordinate is equal to the radius.
Find the equation of Circle L.
The equation of a circle is as follows:
Where 
is the center of the circle and 
is the radius.
Statement I gives us the radius.
Statement II gives us clues to find 
.
So, by using both, we can find the equation. Both are needed.
Recap:
Consider circle L.
I) The radius of L is twice the value of the third prime number.
II) The x-coordinate of L is four times the y-coordinate of L. The y-coordinate is equal to the radius.
Find the equation of circle L.
Using Statement I, the third prime number is 5, so our radius is 10.
Using Statement II, 
, so 
must be 40.
Putting it all into the equation of a circle:
The equation of a circle is as follows:
Where
Statement I gives us the radius.
Statement II gives us clues to find
So, by using both, we can find the equation. Both are needed.
Recap:
Consider circle L.
I) The radius of L is twice the value of the third prime number.
II) The x-coordinate of L is four times the y-coordinate of L. The y-coordinate is equal to the radius.
Find the equation of circle L.
Using Statement I, the third prime number is 5, so our radius is 10.
Using Statement II,
Putting it all into the equation of a circle:
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Circle T is plotted in Euclidean space. Find its equation.
I) The diameter of Circle T is twice the square of the y-coordinate of its center.
II) Circle T is centered at the point 
.
Circle T is plotted in Euclidean space. Find its equation.
I) The diameter of Circle T is twice the square of the y-coordinate of its center.
II) Circle T is centered at the point
The equation of a circle is as follows:
Where 
is the center of the circle and 
is the radius.
Statement I gives us a clue to find the diameter, which can be used to find the radius.
Statement II gives us the center of the circle.
Use Statement II and Statement I to find the radius, then plug it all into the equation to find the equation of Circle T.
Recap:
Circle T is plotted in Euclidean space. Find its equation.
I) The diameter of circle T is twice the square of the y-coordinate of its center.
II) Circle T is centered at the point 
.
Statement II gives us 
.
Use Statement I and Statement II to write the following equation:
So, if the diameter is 18, the radius must be 9. This makes our equation of the circle:
The equation of a circle is as follows:
Where
Statement I gives us a clue to find the diameter, which can be used to find the radius.
Statement II gives us the center of the circle.
Use Statement II and Statement I to find the radius, then plug it all into the equation to find the equation of Circle T.
Recap:
Circle T is plotted in Euclidean space. Find its equation.
I) The diameter of circle T is twice the square of the y-coordinate of its center.
II) Circle T is centered at the point
Statement II gives us
Use Statement I and Statement II to write the following equation:
So, if the diameter is 18, the radius must be 9. This makes our equation of the circle:
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What is the area of the circle 
What is the area of the circle
Circle equations are written in the form 
where 
are the coordinates for the center of the circle. When dealing with the area of a circle, the formula is 
Since, the circle equation already gives us the squared radius, we only need to multiply that number by 
Therefore, the area is 
Circle equations are written in the form
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