Question

Based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. Find an equation of the circle with center at (-3,4) and radius 10

Short answer

(x + 3)^2 + (y - 4)^2 = 100

Step-by-step solution

Recall the Standard Form of a Circle's Equation

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

Substitute the Center Coordinates

Substitute the given center coordinates (-3, 4) into the standard form equation. This gives us:ewline (x + 3)^2 + (y - 4)^2 = r^2

Substitute the Radius

Substitute the given radius 10 into the equation. Simplify the radius to its square: ewline r^2 = 10^2 = 100

Write the Final Equation

Insert the radius squared into the equation from Step 2. The equation of the circle becomes: (x + 3)^2 + (y - 4)^2 = 100

Key concepts

Standard Form of a Circle

In geometry, a circle can be represented using a particular algebraic equation known as the 'standard form of a circle'. This is given by the equation:
\[ (x - h)^2 + (y - k)^2 = r^2 \]
In this equation:
- \(h\) and \(k\) represent the coordinates of the center of the circle.
- \(r\) is the radius of the circle.
The standard form makes it straightforward to understand the position and size of the circle. By examining this equation, one can quickly identify the center and the radius without ambiguity. Make sure to pay attention to the signs; changing them changes the circle's position in the coordinate plane.

Center of a Circle

The center of a circle is a crucial concept in understanding and plotting the circle on a coordinate plane. It tells us the exact middle point of the circle. In our standard form equation:
\[ (x - h)^2 + (y - k)^2 = r^2 \]
the center is represented by the coordinates \( (h, k) \). For instance, in the given exercise, the center is at \( (-3, 4) \). This means that the circle is located 3 units to the left of the origin along the x-axis and 4 units up along the y-axis. By locating the center accurately, you can then determine the circle's entire layout on the graph.

Radius of a Circle

The radius of a circle is the distance from its center to any point on the circle's edge. It plays a vital role in the circle's equation. In the standard form:
\[ (x - h)^2 + (y - k)^2 = r^2 \]
\(r^2\) is the squared length of the radius. For the given problem, the radius provided is 10. Therefore, \(r^2 = 10^2 = 100\). This tells us that every point on the circle is exactly 10 units away from the center \( (-3, 4) \). Knowing the radius helps in sketching the circle accurately, ensuring that it's drawn to scale.