We are given two very important pieces of information.  The first is that the circle exists entirely in the first quadrant, the second is that it intersects both the - and -axis. 

The fact that it is entirely in the first quadrant means that it cannot go past the two axes.  For a circle to intersect the -axis in more than one point, it would necessarily move into another quadrant. Therefore, we can conclude it intersects in exactly one point.

The intersection of the circle with  must also be tangential, since it can only intersect in one point.  We can thus conclude that the circle must have both - and - intercepts equal to 6 and have a center of .

This leaves us with a radius of 6 and an area of:

If the square is inscribed inside the circle, in means the center of the circle is at (1,1). We need to also find the radius of the circle, which happens to be the length from the corner of the square to it's center.

Now use the equation of the circle with the center and .

We get 

We need to expand this equation to  and then complete the square.

This brings us to .

We simplify this to .

Thus the radius is 7.

The radius of the circle is equal to the hypotenuse of a right triangle with sides of lengths 5 and 7.

This radical cannot be reduced further.

The formula for the area of a circle is given by A =πr² .  The problem gives us the endpoints of the diameter of the circle. Using the distance formula, we can find the length of the diameter. Then, because we know that the radius (r) is half the length of the diameter, we can find the length of r. Finally, we can use the formula A =πr² to find the area.

The distance formula is

The distance between the endpoints of the diameter of the circle is:

To find the radius, we divide  d (the length of the diameter) by two.

Then we substitute the value of r into the formula for the area of a circle.

To begin, let us determine the point of intersection of these two lines by setting the equations equal to each other:

To find the y-coordinate, substitute into one of the equations. Let's use :

The center of our circle is therefore .

Now, recall that the general form for a circle with center at  is

For our data, this means that our equation is:

Recall that the equation of a circle is defined as:

Where  is the center of the circle.

Given the data we have, we know that the radius of the circle must be  (half the diameter).  Thus, we know that the equation of the circle in question must be:

Recall that the equation of a circle is defined as:

Where  is the center of the circle.

So, you must first find the center of the circle in question.  This you can do by finding the midpoint of the two points given to us.  Since they represent a diameter of the circle, their midpoint must be the center of the circle.

Recall that the midpoint of two points is found by the equation:

Thus, for our points, we have:

This is:

 or 

Now, the distance between our two points is very easy, as it lies on a horizontal line.  Thus, it is just:

If this is the diameter, the radius of the circle is .  Thus, we know based on our data that our circle's equation must be:

To begin, recall that the equation of a circle is defined as:

, where  is your center point.

Now, for this question it is a bit trickier, for our center point is not a pair of numbers but instead is a set of variables (or at least constants that are not specific numbers). So, for this information, you would know:

None of your answers are in this format (except two that are obviously wrong because of the signs). You need to foil out your groups to find the right answer:

Carefully done, this is: