Problem 42 - Thomas Calculus

Question

In Exercises \(41-46,\) find an equation for the circle with the given center \(C(h, k)\) and radius \(a\) . Then sketch the circle in the \(x y\) -plane. Include the circle's center in your sketch. Also, label the circle's \(x\) - and \(y\) -intercepts, if any, with their coordinate pairs. $$ C(-3,0), \quad a=3 $$

Step-by-Step Solution

Answer

The circle's equation is \((x + 3)^2 + y^2 = 9\). It intersects the axes at (0,0) and (-6,0).

Step 1: Identify Circle Equation Form

The general equation for a circle with center \((h, k)\) and radius \(a\) is given by: \((x - h)^2 + (y - k)^2 = a^2\). Here, \(h = -3\), \(k = 0\), and \(a = 3\).

Step 2: Substitute Values into Circle Equation

Substitute \(h = -3\), \(k = 0\), and \(a = 3\) into the general circle equation: \((x + 3)^2 + (y - 0)^2 = 3^2\), which simplifies to: \((x + 3)^2 + y^2 = 9\).

Step 3: Solve for Intercepts

To find the x-intercepts, set \(y = 0\): \((x + 3)^2 = 9\). Solving gives \(x + 3 = \pm 3\), so \(x = 0\) or \(x = -6\). Thus, the x-intercepts are \((0, 0)\) and \((-6, 0)\). To find the y-intercepts, set \(x = 0\): \((0 + 3)^2 + y^2 = 9\), which simplifies to \(9 + y^2 = 9\), giving \(y = 0\). Therefore, the y-intercept is \((0, 0)\).

Step 4: Sketch the Circle

Draw the circle centered at \((-3, 0)\) with radius 3 on the xy-plane. The circle should pass through \((0, 0)\) and \((-6, 0)\), both x-intercepts, and should also touch the y-axis at \((0, 0)\), the y-intercept.

Key Concepts

circle sketching

Sketching a circle on the coordinate plane is a fun and visual activity that helps you understand circle geometry better. Begin by identifying the center and the radius of the circle. For our exercise, the center is located at \((-3, 0)\) and the radius is \(3\). Use these details to plot the center on the graph. Mark this point clearly so that it guides your sketch.Next, consider drawing a circle around this center point. Utilize a compass to maintain an equal distance from the center to any point on the circle, which helps pinpoint the circle's boundary correctly. The radius of \(3\) means every point on the circle is exactly \(3\) units away from the center.Sketching the circle means visually translating these concepts onto your xy-plane. Make sure your sketch passes through the points that are known intercepts, like \((0, 0)\) and \((-6, 0)\). These should be verified by solving the circle's equation for specific values: making sure the sketch is as accurate as possible, as the circle should touch or pass through these intercept points.

intercepts

Intercepts are vital in understanding where a circle crosses the axes. They reveal the points where geometry meets algebra on the graph. Let's decipher these intercepts for the circle centered at \((-3, 0)\) with radius \(3\).### X-InterceptsWhen finding x-intercepts, the y-coordinate is set to \(0\). Substituting into the equation \((x + 3)^2 + y^2 = 9\), replace \(y\) with \(0\), simplifying the equation to \((x + 3)^2 = 9\). Solving this gives \(x = 0\) and \(-6\). So, the circle meets the x-axis at \((0, 0)\) and \((-6, 0)\).### Y-InterceptTo find the y-intercept, set \(x = 0\). The equation becomes \((3)^2 + y^2 = 9\). Simplified further, it yields \(y = 0\). Therefore, the yellow point \((0, 0)\) is both an x and y-intercept, elucidating how the circle touches the origin of the graph.

circle geometry

Circle geometry may sound complex, but it begins with understanding simple definitions. A circle is a collection of points equidistant from a central point. This aligns perfectly with our exercise.### Equation of a CircleThe fundamental equation for a circle is \((x - h)^2 + (y - k)^2 = a^2\), where - \((h, k)\) is the center of the circle- \(a\) is the radiusFor the given circle, adjusting the general form using \(h = -3\), \(k = 0\), and \(a = 3\), yields \((x + 3)^2 + y^2 = 9\). This equation is the key to unlocking all insights about this circle.### Important Features

• **Center:** Situated at \((-3, 0)\), defining the location of the circle along the plane.
• **Radius:** \(3\) units, indicating the distance from the center to any point on the circle's boundary.
• **Intercepts:** Places where the circle intersects the x-axis or y-axis help understand how it maps onto your graph.

Grasping these core components aids in dissecting any circle equation, providing a deeper comprehension of circle geometry at heart.