Problem 112 - Algebra and Trigonometry

Question

Are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. Write the equation of the circle in standard form: $$ x^{2}+v^{2}-20 x+4 y+55=0 $$

Step-by-Step Solution

Answer

The equation of the circle in standard form is \((x - 10)^2 + (v + 2)^2 = 7^2\).

Step 1: Identify and separate the terms

Group the \(x\) and \(y\) terms together and move the constant to the other side of the equation: \[ x^2 - 20x + v^2 + 4y = -55 \]

Step 2: Complete the square for the \(x\) terms

Take the \(x\) terms: \(x^2 - 20x\). To complete the square, add and subtract \((\frac{-20}{2})^2 = 100\): \[ x^2 - 20x + 100 \]To maintain equation balance, add 100 on the right side as well.

Step 3: Complete the square for the \(y\) terms

Take the \(y\) terms: \(v^2 + 4v\). To complete the square, add and subtract \((\frac{4}{2})^2 = 4\): \[ v^2 + 4v + 4 \]Add 4 to the right side as well.

Step 4: Rewrite the completed squares

Rewrite the equation with the completed squares and balance it:\[ (x - 10)^2 + (v + 2)^2 = -55 + 100 + 4 \]\[ (x - 10)^2 + (v + 2)^2 = 49 \]

Step 5: Write the equation in standard form

The equation of the circle is now in standard form:\[ (x - 10)^2 + (v + 2)^2 = 7^2 \]

Key Concepts

completing the square

Completing the square is a crucial algebraic technique for solving quadratic equations and rewriting them in a more usable form. To complete the square, you need to make a perfect square trinomial from a quadratic expression. Start by taking the quadratic term and the linear term. For example, for the expression \(x^2 - 20x\), you would:

• Divide the coefficient of the linear term by 2: \(-20 / 2 = -10\)
• Square the result: \(-10^2 = 100\)
• Add and subtract this square inside the quadratic expression: \(x^2 - 20x + 100 - 100\)
• Now rewrite it as a perfect square trinomial: \((x - 10)^2 - 100\)

Adding and subtracting the same number does not change the value, allowing you to balance the equation on both sides. Understanding and practicing 'completing the square' simplifies quadratic expressions and is used in various math applications including the equation of a circle.

standard form

The standard form of the equation of a circle is essential for understanding its properties and graphing it. It's concise and tells you everything you need to know about the circle immediately. The standard form is given by: \[(x - h)^2 + (y - k)^2 = r^2\]

In this form:

• \(h\) and \(k\) are the coordinates of the circle's center.
• \(r\) is the radius of the circle.

For example, the equation \((x - 10)^2 + (v + 2)^2 = 49\)\ is in standard form:

• The center of the circle is at \((10, -2)\).
• The radius of the circle is the square root of 49, which is 7.

Recognizing and converting the given general form of an equation into the standard form helps in visualizing and analyzing the circle accurately.

algebra

Algebra lays the foundation for understanding many mathematical concepts, including the equation of a circle.

To manipulate and simplify equations, we use algebraic operations like addition, subtraction, multiplication, and division of terms. These operations allow us to transform the given equation to its required form. In the exercise, we:

• Grouped like terms together to prepare for completing the square.
• Moved constants across the equation for better balance.
• Used coefficients to identify needed additions/subtractions to complete the square.

Each algebraic manipulation step-by-step brings us closer to achieving the desired standard form, crucial for further calculations and interpretations. Practicing these steps strengthens algebraic understanding and problem-solving skills.