Problem 44 - Contemporary Precalculus

Question

Find the \(x\) - and \(y\) -intercepts of the graph of the equation. $$(x+1)^{2}+(y-2)^{2}=4$$

Step-by-Step Solution

Answer

Answer: The x-intercept is (-1,0) and the y-intercepts are (0, 2+√3) and (0, 2-√3).

Step 1: Determine the center and radius

The given equation is: $$(x+1)^{2}+(y-2)^{2}=4$$ Comparing with the standard equation of a circle, we can see that the center is \((-1,2)\) and the radius is \(r=\sqrt{4}=2\).

Step 2: Find the x-intercepts

To find the \(x\)-intercepts, set \(y=0\) and solve for \(x\): $$(x+1)^{2}+(0-2)^{2}=4$$ $$(x+1)^{2}+4=4$$ Now, subtract 4 from both sides to isolate \((x+1)^{2}\): $$(x+1)^{2}=0$$ Square root both sides to get: $$x+1=0$$ $$x=-1$$ So, the \(x\)-intercept is \((-1,0)\).

Step 3: Find the y-intercepts

To find the \(y\)-intercepts, set \(x=0\) and solve for \(y\): $$(0+1)^{2}+(y-2)^{2}=4$$ $$1+(y-2)^{2}=4$$ Now, subtract 1 from both sides to isolate \((y-2)^{2}\): $$(y-2)^{2}=3$$ Square root both sides to get: $$y-2=\pm\sqrt{3}$$ Now, solving for \(y\): $$y=2\pm\sqrt{3}$$ So, the \(y\)-intercepts are \((0, 2+\sqrt{3})\) and \((0, 2-\sqrt{3})\). In conclusion, the \(x\)-intercept is \((-1,0)\) and the \(y\)-intercepts are \((0, 2+\sqrt{3})\) and \((0, 2-\sqrt{3})\).

Key Concepts

Intercepts

When studying circles, finding the intercepts of its equation is a fundamental part of coordinate geometry. An intercept is a point where the circle's graph crosses the axes. In the given equation \[(x+1)^{2}+(y-2)^{2}=4\]we find the intercepts by setting either variable, \(x\) or \(y\), to zero and solving for the other. This equivalently means finding the points on the graph that touch the \(x\)-axis and \(y\)-axis.

• Finding the x-intercept: Set \(y = 0\) and solve for \(x\). The equation turns into \[(x+1)^2 + 4 = 4\] which simplifies to \((x+1)^2 = 0\). Here, \(x = -1\). Thus, the \(x\)-intercept is \((-1,0)\).
• Finding the y-intercepts: Set \(x = 0\) and solve for \(y\). Upon substitution and simplification, the equation becomes \[(y-2)^2 = 3\]. Solving this results in \(y = 2 \pm \sqrt{3}\), giving us the intercepts \((0, 2 + \sqrt{3})\) and \((0, 2 - \sqrt{3})\).

Understanding intercepts allows us to gain insights into the positioning of a circle relative to the coordinate axes.

Standard Form of a Circle

The equation of a circle in its standard form is crucial for determining its properties like the center and radius. A circle's standard form is expressed as \[(x-h)^{2}+(y-k)^{2}=r^{2}\]where \((h, k)\) is the center of the circle and \(r\) is the radius. This form makes it easy to see where the circle is located on a coordinate plane and how large it is.For the example: \[(x+1)^2 +(y-2)^2 = 4\]we can identify the center by comparing the equation to the standard form. Here, \(h = -1\) and \(k = 2\) so the center of this circle is \((-1, 2)\). To determine the radius, we take the square root of the right-hand side of the equation. Since \[r^2 = 4\]then \(r = \sqrt{4} = 2\).This simple rearrangement into standard form offers a visual depiction of the circle, where its position and size are immediately apparent to help with analysis.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, bridges geometry with algebra through graphs and equations. It provides a coordinate system method to explain geometric figures like circles, lines, and parabolas. With coordinate geometry, mathematical problems become visual and involve computations using coordinates on an axis. For instance, the circle's equation \[(x+1)^{2}+(y-2)^{2}=4\]is derived from coordinate geometry principles, enabling us to map the circle on a Cartesian plane.

• Plotting the circle: The center \((-1, 2)\) and radius \(2\) derived from the standard form give the circle its precise location and size on the grid.
• Finding intersections: As demonstrated, setting variables to zero allows for solving the intercepts on the \(x\)-axis and \(y\)-axis.

Coordinate geometry offers powerful tools to dissect and visualize mathematical concepts, providing a deeper understanding of shapes and their interactions with coordinate axes.