Question

In Problems \(25-38,\) (a) find the center \((h, k)\) and radius \(r\) of each circle; \((b)\) graph each circle; \((c)\) find the intercepts, if any. $$ x^{2}+y^{2}=4 $$

Short answer

Center: (0, 0), Radius: 2. Intercepts: (2, 0), (-2, 0), (0, 2), (0, -2).

Step-by-step solution

Identify the Center and Radius

The given equation of the circle is in the standard form: \[ x^2 + y^2 = r^2 \]where \[ (h, k) \] is the center, and \[ r \] is the radius. For the given equation: \[ x^2 + y^2 = 4 \]we can identify that \[ (h, k) = (0, 0) \] and \[ r = \sqrt{4} = 2 \].

Graph the Circle

To graph the circle, plot the center at \[ (0, 0) \]. Then use the radius \[ r = 2 \] to draw a circle around the center. The circle will pass through the points \[ (2, 0) \], \[ (-2, 0) \], \[ (0, 2) \], and \[ (0, -2) \].

Find the Intercepts

To find the intercepts, set \[ y = 0 \] and solve for \[ x \]: \[ x^2 = 4 \] \[ x = 2 \] or \[ x = -2 \]. The x-intercepts are \[ (2, 0) \] and \[ (-2, 0) \]. Similarly, set \[ x = 0 \] and solve for \[ y \]: \[ y^2 = 4 \] \[ y = 2 \] or \[ y = -2 \]. The y-intercepts are \[ (0, 2) \] and \[ (0, -2) \].

Key concepts

center and radius of a circle

To fully understand circle equations, it's crucial to know how to identify the center and the radius of a circle.
The standard equation of a circle is: \( x^2 + y^2 = r^2 \) where \( (h, k) \) represents the center and \( r \) is the radius.

In this equation, there's no \((x - h)\) or \((y - k)\) term, which means both \(h\) and \(k\) are zero. Therefore, the center \((h, k)\) is at \((0, 0)\).
The radius \(r\) is the square root of the number on the right side of the equation.
For the equation \( x^2 + y^2 = 4 \), the radius is \( \sqrt{4} = 2 \).

This means the circle is centered at the origin and has a radius extending from \((0, 0)\) to any point two units away.

graphing circles

Graphing circles becomes straightforward when you know the center and the radius.
First, plot the center on the Cartesian plane. Here, the center is \((0, 0)\).

Next, use the radius to draw the circle. The radius is the distance from the center to any point on the circle.
For a radius of 2, measure 2 units in all directions from the center.
Your circle will touch points \((2, 0)\), \((-2, 0)\), \((0, 2)\), and \((0, -2)\).

Connect these points with a smooth, round curve to complete the circle.
The resulting graph should show a circle centered at the origin and passing through the points identified by the radius.

finding intercepts

Finding intercepts involves seeing where the circle crosses the x-axis and y-axis.

For x-intercepts, set \( y = 0 \) and solve for \( x \).
In the equation \( x^2 + 0^2 = 4 \) which simplifies to \( x^2 = 4 \).
Taking the square root of both sides gives: \( x = 2 \) and \( x = -2 \).
So, the x-intercepts are \((2, 0)\) and \((-2, 0)\).

For y-intercepts, set \( x = 0 \) and solve for \( y \).
The equation becomes \( 0^2 + y^2 = 4 \) which simplifies to \( y^2 = 4 \).
Taking the square root of both sides gives: \( y = 2 \) and \( y = -2 \).
So, the y-intercepts are \((0, 2)\) and \((0, -2)\).

Intercepts are useful in understanding the shape and position of the circle in the coordinate plane.