Step-by-Step Guidance: Circle Equations in College Algebra

Q1. Find the equation of the circle.

Background

Topic: Equation of a Circle

This question tests your understanding of how to write the equation of a circle when given its center and radius.

Key Terms and Formulas

• Standard form of a circle:

  $(x - a)^2 + (y - b)^2 = r^2$

• Center: $(a, b)$

• Radius: $r$

Step-by-Step Guidance

1. Identify the coordinates of the center $(a, b)$ and the radius $r$ from the information or diagram provided.

2. Substitute the values of $a$, $b$, and $r$ into the standard form equation of a circle.

3. Write the equation, making sure to square the radius.

4. Simplify the equation if necessary, but do not expand the squared terms.

Try solving on your own before revealing the answer!

Final Answer:

The center is $(2, 4)$ and the radius is $7$.

The equation is:

$(x - 2)^2 + (y - 4)^2 = 49$

This matches the standard form, with the radius squared ($7^2 = 49$).

Q2. (Image-based) Find the equation of the circle shown.

Background

Topic: Equation of a Circle (Graphical Interpretation)

This question asks you to interpret a graph to determine the center and radius, then write the equation of the circle.

Key Terms and Formulas

• Standard form of a circle:

  $(x - a)^2 + (y - b)^2 = r^2$

• Center: $(a, b)$

• Radius: $r$

Step-by-Step Guidance

1. From the image, identify the center $(a, b)$ and the radius $r$ of the circle.

2. Plug these values into the standard form equation for a circle.

3. Square the radius to complete the equation.

Try solving on your own before revealing the answer!

Final Answer:

The center is $(-2, 4)$ and the radius is $7$.

The equation is:

$(x + 2)^2 + (y - 4)^2 = 49$

Remember, subtracting a negative is the same as adding: $x - (-2) = x + 2$.

Q3. (Image-based) Find the equation of the circle shown.

Background

Topic: Equation of a Circle (Graphical Interpretation)

This question again asks you to use a graph to determine the center and radius, then write the equation of the circle.

Key Terms and Formulas

• Standard form of a circle:

  $(x - a)^2 + (y - b)^2 = r^2$

• Center: $(a, b)$

• Radius: $r$

Step-by-Step Guidance

1. From the image, determine the center $(a, b)$ and the radius $r$.

2. Substitute these values into the standard form equation for a circle.

3. Square the radius to complete the equation.

Try solving on your own before revealing the answer!

Final Answer:

The center is $(-2, -4)$ and the radius is $7$.

The equation is:

$(x + 2)^2 + (y + 4)^2 = 49$

Again, subtracting a negative is the same as adding: $y - (-4) = y + 4$.

Q4. Determine the equation of the circle with the following information: The centre is located at the origin and has a radius of 5.

Background

Topic: Equation of a Circle (Special Case: Center at Origin)

This question tests your ability to write the equation of a circle when the center is at the origin $(0, 0)$.

Key Terms and Formulas

• Standard form of a circle:

  $(x - a)^2 + (y - b)^2 = r^2$

• For center at origin: $a = 0$, $b = 0$

• Radius: $r$

Step-by-Step Guidance

1. Set $a = 0$ and $b = 0$ in the standard form equation.

2. Substitute the given radius $r = 5$ into the equation.

3. Simplify the equation by squaring the radius.

Try solving on your own before revealing the answer!

Final Answer:

The equation is:

$x^2 + y^2 = 25$

This is the standard form for a circle centered at the origin with radius 5.

Q5. Determine the equation of the circle with the following information: The co-ordinates of the diameter (AB) are as follows: A(7, 10), B(11, -2).

Background

Topic: Equation of a Circle (Using Endpoints of Diameter)

This question tests your ability to find the equation of a circle when given the endpoints of its diameter. You need to find the center (midpoint of AB) and the radius (half the distance between A and B).

Key Terms and Formulas

• Midpoint formula:

  $M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$

• Distance formula:

  $D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

• Radius: $r = \frac{D}{2}$

• Standard form of a circle:

  $(x - a)^2 + (y - b)^2 = r^2$

Step-by-Step Guidance

1. Find the midpoint of AB to determine the center of the circle.

2. Calculate the distance between A and B to find the length of the diameter.

3. Divide the diameter by 2 to get the radius.

4. Substitute the center and radius into the standard form equation of a circle.

Try solving on your own before revealing the answer!

Final Answer:

Midpoint: $\left( \frac{7+11}{2}, \frac{10+(-2)}{2} \right) = (9, 4)$

Distance: $\sqrt{(11-7)^2 + ((-2)-10)^2} = \sqrt{16 + 144} = \sqrt{160}$

Radius: $\frac{\sqrt{160}}{2} = 2\sqrt{10}$

Equation: $(x - 9)^2 + (y - 4)^2 = 40$

Q6. Determine the equation of the circle with the following information: With centre point (−2, 7) and radius of 8.

Background

Topic: Equation of a Circle (General Case)

This question tests your ability to write the equation of a circle given any center and radius.

Key Terms and Formulas

• Standard form of a circle:

  $(x - a)^2 + (y - b)^2 = r^2$

• Center: $(a, b)$

• Radius: $r$

Step-by-Step Guidance

1. Identify the center $(a, b)$ and radius $r$ from the information given.

2. Substitute these values into the standard form equation.

3. Square the radius to complete the equation.

Try solving on your own before revealing the answer!

Final Answer:

The equation is:

$(x + 2)^2 + (y - 7)^2 = 64$

Q7. Determine the centre and radius if you are given the equation $(x-3)^2 + (y-6)^2 = 49$.

Background

Topic: Identifying Circle Properties from Standard Form

This question tests your ability to extract the center and radius from the standard form of a circle's equation.

Key Terms and Formulas

• Standard form of a circle:

  $(x - a)^2 + (y - b)^2 = r^2$

• Center: $(a, b)$

• Radius: $r$

Step-by-Step Guidance

1. Compare the given equation to the standard form to identify $a$, $b$, and $r^2$.

2. Extract the values for the center $(a, b)$.

3. Take the square root of the right side to find the radius $r$.

Try solving on your own before revealing the answer!

Final Answer:

Center: $(3, 6)$

Radius: $\sqrt{49} = 7$

Q8. Determine the centre of the circle and radius if you are given the equation $x^2 + 2x + y^2 - 4y = 9$ (Complete the square).

Background

Topic: Completing the Square to Find Circle Properties

This question tests your ability to rewrite a general quadratic equation in standard form by completing the square, so you can identify the center and radius.

Key Terms and Formulas

• Completing the square: A method to rewrite $x^2 + bx$ as $(x + d)^2 + c$.

• Standard form of a circle:

  $(x - a)^2 + (y - b)^2 = r^2$

Step-by-Step Guidance

1. Group the $x$ and $y$ terms: $x^2 + 2x$ and $y^2 - 4y$.

2. Complete the square for both $x$ and $y$ terms:

   • For $x^2 + 2x$, add and subtract $(2/2)^2 = 1$.

   • For $y^2 - 4y$, add and subtract $(-4/2)^2 = 4$.

3. Rewrite the equation in standard form and move constants to the right side.

4. Identify the center $(a, b)$ and radius $r$ from the completed square form.

Try solving on your own before revealing the answer!

Final Answer:

After completing the square:

$(x + 1)^2 + (y - 2)^2 = 12$

Center: $(-1, 2)$

Radius: $\sqrt{12} = 2\sqrt{3}$