Calculating the equation of a circle - GMAT Quantitative

The equation for a circle is $(x-a)^{2}$$+(y-b)^{2}$ = $r^{2}$, where (a, b) is the center and r is the radius. In our equation, a = –3, b = 0, and r = $\sqrt{2}$. Then the equation describes a circle with a center at (–3, 0) and a radius of $\sqrt{2}$.

The equation of a circle with center and radius is

The center is the midpoint of any diameter, so to find the center, we use the midpoint formula:

The center is (7,3). The radius is the distance between (7,3) and (10,1), so we use the distance formula:

So , and the equation of the circle is

The distance of the two points is .

So 10 is the radius of the circle. Then we can calculate the area:

.

The equation of a circle with center at the origin is

where is the radius of the circle. The area of the circle is .

Since the area of the circle in the question is , we can solve for :

The equation is

The equation of a circle in gneral form is:

In this case, our radius is 6, because our circle touches the y-axis once at the point (0,-5). This makes our radius equal to the absolute value of the x-coordinate of the center of our circle. Eliminate anything that doesn't have a 36.

Then, because our (h,k) are already negative, they change the signs to positive within the parentheses making our answer:

The general form for an equation of a circle is as follows:

Where r is the radius, and (h,k) are the coordinates of the center of the circle.

To begin, let's find the radius using distance formula. Because LK passes through the center of the circle and goes from the outer edge of the circle to the other side, we can say that LK is our diameter.

Use distance formula to find the length of LK.

Plug in our points and simplify:

That is our diameter, so our radius will be half of 13.15, or 6.575. This rounds to 6.58

Next, we can use midpoint formula to find the center of circle Q. Midpoint formula is:

Plug in and simplify to find our midpoint

Put it all together to get:

To solve this problem, remember that the general formula for a circle with center and radius is:

Therefore,

To solve, simply use the formula given center at and radius of .

A circle with its center at the origin has as its equation

.

Since the equation of the outer circle is , then, for this circle,

,

and its area is .

The area of the region between the circles is , so the inner circle has area

.

If is the radius of the inner circle, then its area is

This makes , and the equation of the inner circle

or

A circle with its center at the origin has as its equation

.

Since the equation of the inner circle is , then, for this circle,

,

and its area is .

The area of the region between the circles is , so the outer circle has area

.

If is the radius of the outer circle, then its area is

This makes , and the equation of the outer circle

or

A circle with its center at the origin has as its equation

.

Let and be the radii of the larger and smaller circles, respectively.

The larger circle has equation , so . The area of a circle is equal to times the square of the radius, so the area of the larger circle is .

Similarly, the area of the smaller circle, whose equation is , is .

The area of the region between them is the difference, which is .

The equation of a circle centered at the origin is

where is the radius of the circle. Since the equation of the outer circle is

,

,

and the radius is the square root of this:

.

The circumference of this circle is times this, or

.

The circumference of the larger circle is twice this, or ; divide this by to get the radius of the larger circle, which is

.

Consequently, .

The equation of the larger circle is

, or

.

The equation of a circle centered at the origin is

where is the radius of the circle. Since the equation of the outer circle is

,

,

and the area is

.

The area of the larger circle is twice this, or ; that is,

,

and the equation of that outer circle is

.

Let be the radius of the inner circle. The area of the inner circle is ; the outer circle has area five times this, or ; the region between them has area equal to the difference of these quantities, or

This is equal to , so

A circle with its center at the origin has as its equation

,

so the inner circle has as its equation

.

The general form for an equation of a circle is as follows:

Where r is the radius, and (h,k) are the coordinates of the center of the circle.

To begin, let's find the radius using distance formula. Because LK passes through the center of the circle and goes from the outer edge of the circle to the other side, we can say that LK is our diameter.

Use distance formula to find the length of LK.

Plug in our points and simplify:

That is our diameter, so our radius will be half of 13.15, or 6.575. This rounds to 6.58

Next, we can use midpoint formula to find the center of circle Q. Midpoint formula is:

Plug in and simplify to find our midpoint

Put it all together to get:

a and b are the right-angle sides of a triangle, and the radius of the circle is the hypotenuse of the triangle. From the Pythagorean Theorem we would know that $a^{2}$$+b^{2}$$=r^{2}$$=5^{2}$=25.

The equation for a circle is $(x-a)^{2}$$+(y-b)^{2}$ = $r^{2}$, where (a, b) is the center and r is the radius. In our equation, a = –3, b = 0, and r = $\sqrt{2}$. Then the equation describes a circle with a center at (–3, 0) and a radius of $\sqrt{2}$.

The equation of a circle with center and radius is

The center is the midpoint of any diameter, so to find the center, we use the midpoint formula:

The center is (7,3). The radius is the distance between (7,3) and (10,1), so we use the distance formula:

So , and the equation of the circle is

The distance of the two points is .

So 10 is the radius of the circle. Then we can calculate the area:

.