The image shows a math problem related to the equation of a circle. Here is the content extracted from the image:

1. Determine the coordinates of the center and radius of the circle from the equation given below: $x^2 + y^2 = 72$

2. Find the standard equation of the circle which satisfies the given conditions:

   • Center at $(3, 7)$
   • Tangent to the x-axis
   • Another condition: $x^2 + y^2 = 72$

Solution to Part 1:

The general equation of a circle in standard form is: $(x - h)^2 + (y - k)^2 = r^2$ where $(h, k)$ is the center of the circle and $r$ is the radius.

The given equation is: $x^2 + y^2 = 72$

This equation can be rewritten in the standard form as: $(x - 0)^2 + (y - 0)^2 = 72$

From this, we see that:

• The center of the circle is at $(0, 0)$.
• The radius $r$ is $\sqrt{72} = 6\sqrt{2}$.

Solution to Part 2:

Now, for the second part:

• The center of the circle is given as $(3, 7)$.
• The circle is tangent to the x-axis.

For the circle to be tangent to the x-axis, the distance from the center $(3, 7)$ to the x-axis must be equal to the radius $r$. The distance from a point $(x_1, y_1)$ to the x-axis is simply the absolute value of its y-coordinate, $|y_1|$.

Here, the y-coordinate of the center is 7, so the radius $r = 7$.

Now, the equation of the circle in standard form is: $(x - 3)^2 + (y - 7)^2 = 7^2$

Simplifying, we get: $(x - 3)^2 + (y - 7)^2 = 49$

Summary:

1. Coordinates of the center and radius of the first circle:

   • Center: $(0, 0)$
   • Radius: $6\sqrt{2}$

2. Equation of the second circle:

   • Center: $(3, 7)$
   • Radius: $7$
   • Equation: $(x - 3)^2 + (y - 7)^2 = 49$

Would you like more details or have any questions?

Related Questions:

1. How do you find the equation of a circle tangent to both axes?
2. How is the distance from a point to a line (such as the x-axis) calculated?
3. What is the geometric interpretation of a circle being tangent to a line?
4. Can there be more than one circle that satisfies a given set of conditions?
5. How do we determine if two circles are tangent to each other?

Tip:

Remember, a circle tangent to the x-axis has its radius equal to the absolute value of its y-coordinate.