Problem 15 - Calculus Single Variable

Question

The Cartesian equation of a circle is given. Sketch the circle and specify its center and radius. \(4 x^{2}+4 y^{2}+8 y-16 x=0\)

Step-by-Step Solution

Answer

The circle has a center at (2, -1) and a radius of \(\sqrt{5}\).

Step 1: Start with the Given Equation

The given equation of the circle is \(4x^2 + 4y^2 + 8y - 16x = 0\). Our goal is to rewrite it in a form that reveals the center and radius of the circle.

Step 2: Simplify the Equation

Divide the entire equation by 4 to simplify: \(x^2 + y^2 + 2y - 4x = 0\). This makes it easier to complete the square in the next step.

Step 3: Rearrange and Complete the Square for x

Rearrange the terms for x: \((x^2 - 4x)\). Complete the square by adding and subtracting \(4\) inside the expression: \((x - 2)^2 - 4\).

Step 4: Rearrange and Complete the Square for y

Rearrange the terms for y: \((y^2 + 2y)\). Complete the square by adding and subtracting \(1\) inside the expression: \((y + 1)^2 - 1\).

Step 5: Rewrite the Equation

Substitute the completed squares back into the equation: \((x - 2)^2 - 4 + (y + 1)^2 - 1 = 0\). Simplify by adding \(4 + 1 = 5\) to both sides: \((x - 2)^2 + (y + 1)^2 = 5\).

Step 6: Identify Center and Radius

The equation \((x - 2)^2 + (y + 1)^2 = 5\) reveals that the circle has a center at \((2, -1)\) and a radius of \(\sqrt{5}\).

Step 7: Sketch the Circle

Draw a Cartesian plane and plot the center of the circle at \((2, -1)\). Use the radius \(\sqrt{5}\), which is approximately 2.24, to draw the circle. Since the radius is less than 3, ensure the circle stays within a reasonable scale.

Key Concepts

Completing the Square

Completing the square is a helpful method in algebra that can transform quadratic expressions into a manageable form. For circle equations, this method is crucial to identify the circle's center and radius by rewriting the circle's equation in a standard form. When dealing with circle equations like \(x^2 + y^2 + 2y - 4x = 0\), we use completing the square separately for both the \(x\) and \(y\) terms.

• To complete the square for \(x\) in \(x^2 - 4x\), you find the middle value of the linear term (-4), halve it, and square it. That's −2 squared, or 4.
• Apply the same process to \(y\) in \(y^2 + 2y\), which gives us a square of 1 since halving 2 gives 1.

This gives us \[(x - 2)^2 - 4\] and \[(y + 1)^2 - 1\].Add those adjustments back in to keep the equation balanced, and you'll get a simplified circle equation ready for interpretation.

Cartesian Plane

The Cartesian plane, also known as the xy-plane, is a two-dimensional graph used in mathematics to display and explore geometric figures. It comprises two perpendicular number lines: the horizontal line (x-axis) and the vertical line (y-axis). Together, they form a grid that can represent points, lines, and curves — such as circles.To sketch a circle given by its equation, like in our exercise, you plot the center of the circle and trace out a curve equal to the radius distance from the center in all directions. Given the center at \((2, -1)\) and radius \(\sqrt{5}\),

• First plot the center on the plane.
• Then mark the radius outwards equidistant from this point in every direction.

Visualizing this on the Cartesian plane, you essentially draw a perfect loop around the plotted center, verifying the dimensions against the given radius.

Center of a Circle

In the context of circle equations, the center is a pivotal point that defines where the circle is positioned on a Cartesian plane. Once circle equations are expressed in the standard form \((x - h)^2 + (y - k)^2 = r^2\),the coordinates \((h, k)\)represent the center.For the given equation \((x - 2)^2 + (y + 1)^2 = 5\), we identify the center as \((2, -1)\).

• The value \(h\) is taken from \(x - h\), which is \(2\).
• The value \(k\) is from \(y - k\), but represented with its opposite sign since the equation has a "+k" form, giving us \(-1\).
• Plotting this point precisely allows for an accurate depiction of the circle in its relative position on the plane.

Radius of a Circle

The radius of a circle is the distance from its center to any point on its edge, and it appears in the circle's equation as \((x - h)^2 + (y - k)^2 = r^2\).The term on the right typically represents the radius squared, \(r^2\). To find the radius from our equation \((x - 2)^2 + (y + 1)^2 = 5\), we simply take the square root of 5 because it’s in the form \(r^2 = 5\), yielding \(r = \sqrt{5}\). This gives a numerical approximation of about 2.24.

• Always remember the radius is the same in every direction from the center, keeping the circle equidistant and perfectly round.
• The understanding of radius assists in accurately sketching the circle on the Cartesian plane.