Problem 37 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the \(x\) - and \(y\) -intercepts of the graph of the circle. $$(x-6)^{2}+(y+3)^{2}=16$$

Step-by-Step Solution

Answer

The \(x\)-intercepts of the graph are \(x=3, 9\) and the \(y\)-intercepts are \(y=-7, 1\).

Step 1: Find the x-intercepts

Set \(y=0\) in the equation to find the \(x\)-intercepts. This gives \((x-6)^2+(0+3)^2=16\), which simplifies to \((x-6)^2+9=16\). Solving this shows that \(x=3, 9\) are the \(x\)-intercepts of the graph.

Step 2: Find the y-intercepts

Set \(x=0\) in the equation to find the \(y\)-intercepts. This gives \((0-6)^2+(y+3)^2=16\), simplifying to \(36+(y+3)^2=16\). Solving this yields \(y=-7, 1\) as the \(y\)-intercepts of the graph.

Key Concepts

Algebraic Circles

In coordinate geometry, an algebraic circle is defined by an equation of the form
\((x - h)^2 + (y - k)^2 = r^2\), where \((h, k)\) is the center of the circle and \(r\) is its radius. By squaring the radius and setting both \(x\) and \(y\) to particular values, one can determine specific points on the circle's perimeter, namely its intercepts with the axes. The beauty of algebraic circles lies in their symmetry, which is why the process to find the intercepts with the \(x\)-axis (where \(y = 0\)) or the \(y\)-axis (where \(x = 0\)) is systematic and straightforward.

• Center of the Circle: \((h, k)\)
• Radius of the Circle: \(r\)
• Standard Form Equation: \((x - h)^2 + (y - k)^2 = r^2\)

Solving Circle Equations

Solving equations of circles typically involves manipulating the equation to understand its properties, including center, radius, and intercepts. When given a circle's equation, like \((x - 6)^2 + (y + 3)^2 = 16\), it's important to remember that solving for intercepts involves making one variable zero and solving for the other. This process isolates the variable of interest and makes it much simpler to deal with. For example, to find the \(x\)-intercepts, set \(y = 0\) and solve for \(x\). For the \(y\)-intercepts, do the inverse by setting \(x = 0\) and solve for \(y\). Throughout this process, knowledge of how to expand squares and solve quadratic equations is crucial.

Key Steps in Solving for Intercepts:

• To find \(x\)-intercepts, set \(y=0\) and solve for \(x\)
• To find \(y\)-intercepts, set \(x=0\) and solve for \(y\)

Graphing Circles

Graphing a circle in coordinate geometry involves plotting its center at \((h, k)\) and using the radius \(r\) to determine the extent of the circle around the center. The intercepts found by solving the circle's equation provide critical points that you can mark on your graph, showing where the circle crosses the axes. In the case of the equation \((x - 6)^2 + (y + 3)^2 = 16\), after finding the \(x\) and \(y\) intercepts (\(x = 3, 9\) and \(y = -7, 1\) respectively), plot these points to help shape the circle on the coordinate plane.

Important Aspects of Graphing:

• Plot the center \((h, k)\)
• Use radius \(r\) to draw the circle from the center
• Mark the \(x\)- and \(y\)-intercepts as guide points

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, employs the principles of algebra to geometric problems through a coordinate system, usually the Cartesian plane. Circles are just one of many geometric shapes that can be represented algebraically in this system. To master problems involving circles on the coordinate plane, one should be comfortable with plotting points, understanding the meaning of the coordinates and axes, and performing algebraic manipulations.
By understanding the interactions between algebra and geometry, students can solve complex problems by breaking them down into comprehensible steps—finding intercepts of a circle being one such task. Mastery of coordinate geometry is essential for interpreting equations graphically and translating geometric features into algebraic language and vice versa.

Key Concepts in Coordinate Geometry:

• Cartesian coordinate system
• Plotting points (intercepts, centers, etc.)
• Interrelationship of algebra and geometry