By graphing the given equations, we can determine the number of points of intersection between the circle and the line. To do so, we need to graph both figures. Let's consider one of them at a time.
Graphing the Circle
Let's start by recalling the standard equation of a circle.
(x- h)^2+(y- k)^2= r^2
Here, the center is the point ( h, k) and the radius is r. We will rewrite the given equation to match this form, and then we can identify the center and the radius.
(x-1)^2+y^2=4
⇕
(x- 1)^2+(y- 0)^2= 2^2
The center of the circle is the point ( 1, 0), and its radius is 2. We can graph the circle using this information.
Graphing the Line
To graph the line, we will need its equation to be in slope-intercept form to help us identify the slope m and y-intercept b. Fortunately, our equation is already in slope-intercept form. Let's rewrite it a bit, so that the slope and y-intercept are more visible.
y=x+1 ⇔ y= 1x+ 1
We can see that the slope is 1 and the y-intercept is (0, 1). To graph this equation, we will start by plotting its y-intercept. Then, we will use the slope to determine another point that satisfies the equation, and connect the points with a line.
Finding the Points of Intersection
We can see that the circle and the line intersect at exactly two points. Let's identify them.
