How to find the equation of a circle - SSAT Upper Level Quantitative
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Give the equation of the above circle.
Give the equation of the above circle.
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A circle with center 
and radius 
has equation
The circle has center 
and radius 4, so substitute:
A circle with center
The circle has center
Give the equation of the above circle.
Give the equation of the above circle.
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A circle with center 
and radius 
has equation
The circle has center 
and radius 5, so substitute:
A circle with center
The circle has center
A circle on the coordinate plane has a diameter whose endpoints are 
and 
. Give its equation.
A circle on the coordinate plane has a diameter whose endpoints are
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A circle with center 
and radius 
has equation
The midpoint of a diameter of the circle is its center, so use the midpoint formula to find this:
Therefore, 
and 
The radus is the distance between the center and one endpoint, so take advantage of the distance formula using 
and 
. We will concern ourcelves with finding the square of the radius 
:
Substitute:
A circle with center
The midpoint of a diameter of the circle is its center, so use the midpoint formula to find this:
Therefore,
The radus is the distance between the center and one endpoint, so take advantage of the distance formula using
Substitute:
A circle on the coordinate plane has a diameter whose endpoints are 
and 
. Give its equation.
A circle on the coordinate plane has a diameter whose endpoints are
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A circle with center and radius has equation
The midpoint of a diameter of the circle is its center, so use the midpoint formula to find this:
Therefore, 
and 
.
The radus is the distance between the center and one endpoint, so take advantage of the distance formula using 
and 
. We will concern ourcelves with finding the square of the radius :
Substitute:
Expand:
A circle with center
The midpoint of a diameter of the circle is its center, so use the midpoint formula to find this:
Therefore,
The radus is the distance between the center and one endpoint, so take advantage of the distance formula using
Substitute:
Expand:
A circle on the coordinate plane has center 
and circumference 
. Give its equation.
A circle on the coordinate plane has center
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A circle with center and radius has equation
The center is 
, so 
.
To find 
, use the circumference formula:
Substitute:
A circle with center
The center is
To find
Substitute:
A circle on the coordinate plane has center 
and area 
. Give its equation.
A circle on the coordinate plane has center
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A circle with center and radius has the equation
The center is 
, so 
.
The area is 
, so to find 
, use the area formula:
The equation of the line is therefore:
A circle with center
The center is
The area is
The equation of the line is therefore:
What is the equation of a circle that has its center at 
and has a radius of 
?
What is the equation of a circle that has its center at
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The general equation of a circle with center 
and radius 
is:
Now, plug in the values given by the question:
The general equation of a circle with center
Now, plug in the values given by the question:
If the center of a circle with a diameter of 5 is located at 
, what is the equation of the circle?
If the center of a circle with a diameter of 5 is located at
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Write the formula for the equation of a circle with a given point, 
.
The radius of the circle is half the diameter, or 
.
Substitute all the values into the formula and simplify.
Write the formula for the equation of a circle with a given point,
The radius of the circle is half the diameter, or
Substitute all the values into the formula and simplify.
Give the circumference of the circle on the coordinate plane whose equation is
Give the circumference of the circle on the coordinate plane whose equation is
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The standard form of the equation of a circle is
where 
is the radius of the circle.
We can rewrite the equation we are given, which is in general form, in this standard form as follows:
Complete the squares. Since 
and 
, we do this as follows:
, so 
, and the circumference of the circle is
The standard form of the equation of a circle is
where
We can rewrite the equation we are given, which is in general form, in this standard form as follows:
Complete the squares. Since
Which of the following is the equation of a circle with center at the origin and circumference 
?
Which of the following is the equation of a circle with center at the origin and circumference
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The standard form of the equation of a circle is
,
where the center is 
and the radius is 
.
The center of the circle is the origin, so 
.
The equation will be
for some 
.
The circumference of the circle is 
, so
The equation is 
, which is not among the responses.
The standard form of the equation of a circle is
where the center is
The center of the circle is the origin, so
The equation will be
for some
The circumference of the circle is
The equation is
Which of the following is the equation of a circle with center at the origin and area 
?
Which of the following is the equation of a circle with center at the origin and area
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The standard form of the equation of a circle is
,
where the center is 
and the radius is 
.
The center of the circle is the origin, so 
, and the equation is
for some 
.
The area of the circle is 
, so
We need go no further; we can substitute to get the equation 
.
The standard form of the equation of a circle is
where the center is
The center of the circle is the origin, so
for some
The area of the circle is
We need go no further; we can substitute to get the equation
A square on the coordinate plane has as its vertices the points 
. Give the equation of a circle circumscribed about the square.
A square on the coordinate plane has as its vertices the points
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Below is the figure with the circle and square in question:
The center of the inscribed circle coincides with that of the square, which is the point 
. Its diameter is the length of a diagonal of the square, which is 
times the sidelength 6 of the square - this is 
. Its radius is, consequently, half this, or 
. Therefore, in the standard form of the equation,
,
substitute 
and 
.
Below is the figure with the circle and square in question:
The center of the inscribed circle coincides with that of the square, which is the point
substitute
A square on the coordinate plane has as its vertices the points 
. Give the equation of a circle inscribed in the square.
A square on the coordinate plane has as its vertices the points
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Below is the figure with the circle and square in question:
The center of the inscribed circle coincides with that of the square, which is the point 
. Its diameter is equal to the sidelength of the square, which is 8, so, consequently, its radius is half this, or 4. Therefore, in the standard form of the equation,
,
substitute 
and 
.
Below is the figure with the circle and square in question:
The center of the inscribed circle coincides with that of the square, which is the point
substitute
Give the area of the circle on the coordinate plane whose equation is
.
Give the area of the circle on the coordinate plane whose equation is
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The standard form of the equation of a circle is
where 
is the radius of the circle.
We can rewrite the equation we are given, which is in general form, in this standard form as follows:
Complete the squares. Since 
and 
, we do this as follows:
, and the area of the circle is
The standard form of the equation of a circle is
where
We can rewrite the equation we are given, which is in general form, in this standard form as follows:
Complete the squares. Since
Give the equation of the above circle.
Give the equation of the above circle.
Tap to see back →
A circle with center and radius has equation
The circle has center and radius 4, so substitute:
A circle with center
The circle has center
Give the equation of the above circle.
Give the equation of the above circle.
Tap to see back →
A circle with center and radius has equation
The circle has center and radius 5, so substitute:
A circle with center
The circle has center
A circle on the coordinate plane has a diameter whose endpoints are and . Give its equation.
A circle on the coordinate plane has a diameter whose endpoints are
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A circle with center and radius has equation
The midpoint of a diameter of the circle is its center, so use the midpoint formula to find this:
Therefore, and
The radus is the distance between the center and one endpoint, so take advantage of the distance formula using and . We will concern ourcelves with finding the square of the radius :
Substitute:
A circle with center
The midpoint of a diameter of the circle is its center, so use the midpoint formula to find this:
Therefore,
The radus is the distance between the center and one endpoint, so take advantage of the distance formula using
Substitute:
A circle on the coordinate plane has a diameter whose endpoints are and . Give its equation.
A circle on the coordinate plane has a diameter whose endpoints are
Tap to see back →
A circle with center and radius has equation
The midpoint of a diameter of the circle is its center, so use the midpoint formula to find this:
Therefore, and .
The radus is the distance between the center and one endpoint, so take advantage of the distance formula using and . We will concern ourcelves with finding the square of the radius :
Substitute:
Expand:
A circle with center
The midpoint of a diameter of the circle is its center, so use the midpoint formula to find this:
Therefore,
The radus is the distance between the center and one endpoint, so take advantage of the distance formula using
Substitute:
Expand:
A circle on the coordinate plane has center and circumference . Give its equation.
A circle on the coordinate plane has center
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A circle with center and radius has equation
The center is , so .
To find , use the circumference formula:
Substitute:
A circle with center
The center is
To find
Substitute:
A circle on the coordinate plane has center and area . Give its equation.
A circle on the coordinate plane has center
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A circle with center and radius has the equation
The center is , so .
The area is , so to find , use the area formula:
The equation of the line is therefore:
A circle with center
The center is
The area is
The equation of the line is therefore: