Review on Circles

Summary

This video provides a review of key concepts related to circles, specifically focusing on the standard form equation of a circle. The presenter explains how to find the equation of a circle given its center and radius, determine the center and radius from an equation, and solve problems involving points on the circle or endpoints of a diameter. The video also covers how to convert a circle's general form to its standard form. This comprehensive review helps students solve various problems involving circles efficiently.

Takeaways

• 🔵 The video provides a review of circles, starting with a discussion on their standard form, commonly encountered in precalculus.
• 🟢 The standard form of a circle is given as (x - h)² + (y - k)² = r², where (h, k) is the center and r is the radius.
• 🔍 The first problem solves for the standard form of a circle given a center of (-1, 3) and radius of 9, resulting in the equation (x + 1)² + (y - 3)² = 81.
• 🧮 When the center of the circle is at the origin (0, 0) and the radius is 3, the standard form is x² + y² = 9.
• 🔄 To find the center and radius from a given standard form, the signs of h and k are reversed, and the radius is calculated from the right side of the equation.
• 📝 Example: Given the equation (x + 9)² + (y - 5)² = 16, the center is (-9, 5) and the radius is 4.
• 🧩 A problem involving a center and a point on the circle is solved by finding the radius through the distance formula, then using it in the standard circle equation.
• 📏 Midpoint formulas are used to find the center when two endpoints of the diameter are given, helping solve for the circle's equation.
• 🔧 When converting from general form to standard form, the process involves completing the square for both x and y terms.
• ✅ Once the equation is simplified, you can determine the center and radius, as demonstrated in the final problem example.

Q & A

What is the standard form equation of a circle?

The standard form equation of a circle is (x - h)² + (y - k)² = r², where (h, k) is the center of the circle, and r is the radius.

How do you find the standard form of a circle given the center (-1, 3) and radius 9?

To find the standard form, substitute the center coordinates and radius into the equation: (x - (-1))² + (y - 3)² = 9², which simplifies to (x + 1)² + (y - 3)² = 81.

What is the standard form equation of a circle centered at the origin with a radius of 3?

The standard form equation for a circle centered at the origin (0, 0) with a radius of 3 is x² + y² = 3², which simplifies to x² + y² = 9.

How do you determine the center and radius from a given standard form equation of a circle?

For an equation like (x + 9)² + (y - 5)² = 16, the center is found by reversing the signs of the constants inside the parentheses, giving the center (-9, 5). The radius is the square root of the number on the right side of the equation, so the radius is √16 = 4.

How do you find the radius of a circle when given the center (3, 4) and a point on the circle (-2, 4)?

To find the radius, use the distance formula between the center and the point on the circle: √((-2 - 3)² + (4 - 4)²) = √((-5)² + 0) = √25 = 5. Thus, the radius is 5.

What is the midpoint formula, and how is it used to find the center of a circle when given the endpoints of its diameter?

The midpoint formula is ((x₁ + x₂) / 2, (y₁ + y₂) / 2). It’s used to find the center of a circle by averaging the x and y coordinates of the endpoints of the diameter.

How do you find the radius when given two points on the diameter of a circle?

First, find the center using the midpoint formula. Then, calculate the radius by finding the distance between the center and one of the points using the distance formula.

What is the process to convert a general form equation of a circle to the standard form?

The process involves four steps: 1) Group the x and y terms, 2) Complete the square for both x and y, 3) Add the necessary constants to both sides of the equation, and 4) Factor and simplify.

How do you complete the square for the expression x² - 10x to help convert a general equation into standard form?

Take the coefficient of x (which is -10), divide it by 2 to get -5, and then square it to get 25. Add 25 to both sides of the equation.

What is the radius of a circle if the standard form is (x - 5)² + (y - 7)² = 81?

The radius is the square root of 81, which is 9.