Question

Find the equation of the circle satisfying the given conditions. Center $(3,4)$ and tangent to $x$ -axis

Short answer

Equation: $(x-3{)}^2+(y-4{)}^2=16$.

Step-by-step solution

Understand the properties of the circle

A circle is defined by its center and its radius. The general equation of a circle centered at $(h,k)$ is $(x-h{)}^2+(y-k{)}^2=r^2$ where $r$ is the radius of the circle.

Identify given information

We are given that the center of the circle is $(3,4)$. Thus, $h=3$ and $k=4$. The x-axis serves as a tangent to the circle.

Determine the radius

Since the circle is tangent to the x-axis, the radius is the vertical distance from the center of the circle to the x-axis. The y-coordinate of the center is $4$, so the radius $r=4$.

Write the equation using the circle formula

Substitute $h=3$, $k=4$, and $r=4$ into the circle equation to get $(x-3{)}^2+(y-4{)}^2=4^2$. Therefore, the equation of the circle is: $(x-3{)}^2+(y-4{)}^2=16$.

Key concepts

Circle Properties

A circle is a unique geometric shape defined by its center and radius. The center of a circle is the point from which every point on the circumference is equidistant. This distance is known as the radius. Understanding the circle's center and radius helps us grasp many of its properties and aids in forming the circle's equation.

Some essential properties of circles include:

• The circumference, which is the distance around the circle. It is calculated using the formula $2\pi r$, where $r$ is the radius.
• The area, which is the space enclosed by the circle, determined by the formula $\pi r^2$.
• All radii within a circle are equal in length.
• A diameter is twice the length of the radius and passes directly through the center.

These fundamental properties are crucial for solving circle-related problems and understanding more complex concepts in geometry.

Radius Determination

Determining the radius of a circle is a vital step in finding its equation or understanding its geometry. When you know the circle's center and its relationship with the coordinate axes or any line, you can calculate the radius easily.

In the provided exercise, the circle is tangent to the $x$-axis. This tangency means that the circle just touches the $x$-axis without crossing it. Since it's tangent at this point, the radius is exactly equal to the vertical distance from the center of the circle to the $x$-axis.

With a center at $(3,4)$, the y-coordinate directly indicates the radius, $r=4$. This is because the vertical distance from the center to the $x$-axis (which is $y=0$) is simply $4$.

Circle Equation

The equation of a circle is derived using its center and radius. The standard formula for a circle centered at $(h,k)$ with radius $r$ is:

• $(x-h{)}^2+(y-k{)}^2=r^2$

This equation allows us to represent any circle on a coordinate plane.

In our specific problem, we have a circle with center $(3,4)$ and radius $4$. Substituting these values into the standard equation, we get:

• $(x-3{)}^2+(y-4{)}^2=4^2$

This simplifies to:

• $(x-3{)}^2+(y-4{)}^2=16$

Thus, this is the equation of the circle, which can be used to analyze and graph its points and understand its spatial relationships.

Circle Tangent to Axis

When a circle is tangent to an axis, it means one of its sides touches the axis at exactly one point, making it a unique position for the circle on the coordinate plane. This point of tangency shows a special geometric relationship between the circle and the axis.

In our exercise, the circle is tangent to the $x$-axis. This conveys that the bottommost part of the circle touches the $x$-axis, while the rest of the circle remains above. This tangency has implications:

• The radius is equal to the y-coordinate of the circle's center, because that's the distance from the center directly to the axis.
• It helps to quickly identify geometrical changes like intersection or remain independent of the neighboring shapes.

This concept is useful in many applications, as it quickly determines circle placement and properties without extensive calculation.