2.4.4 Understanding, using the equation circle centered

• What is the relationship between the area and radius of a circle?

  • The area of a circle is proportional to the square of its radius
  • The formula for the area of a circle is $A =$$\pi r^2$
  • As the radius increases, the area increases exponentially
• What is the equation of a circle centered at the origin?

  $x^{2} +$$y^{2} =$$r^{2}$
• What is the middle point of a circle called?

  Origin
• What is the formula for the area of a circle?

  $A =$$\pi r^2$
• What is the equation used to check if a point lies inside, on, or outside a circle centered at the origin?

  $x^{2} +$$y^{2} =$$r^{2}$
• If a point $(3, 4)$ lies on the circle, what is the value of $3^{2} +$$4^{2}$?


  $25$
• What is the formula for the circumference of a circle?

  $C =$$2\pi r$
• Where does the point $(3, 0)$ lie relative to the circle $x^{2} +$$y^{2} =$$9$?


  On the circle
• Steps to find the radius of a circle using the equation $x^{2} +$$y^{2} =$$r^{2}$


  1️⃣ Substitute $x$ and $y$ values into the equation

  2️⃣ Solve for $r^{2}$

  3️⃣ Take the square root of $r^{2}$ to find $r$
• What is the formula that relates the radius of a circle to its area?

  $A =$$\pi r^{2}$
• What is the equation used to find the radius of a circle when given a point on the circle?

  $x^{2} +$$y^{2} =$$r^{2}$
• What are the coordinates of the center of the circle?

  (-1, 3)
• What is the standard form of a circle equation?

  $(x - a)^{2} +$$(y - b)^{2} =$$r^{2}$
• How can you calculate the rate of change of the area of a circle with respect to its radius?

  • The rate of change of the area with respect to the radius is given by the derivative:

  $\frac{dA}{dr} =$$2\pi r$

  • This represents the instantaneous rate of change of the area as the radius changes
• What do you solve for after substituting the values of $x$ and $y$ into the equation?


  $r^{2}$
• Match the condition with the point's location:

  $x^{2} +$$y^{2} < r^{2}$ ↔️ Inside the circle

  $x^{2} +$$y^{2} =$$r^{2}$ ↔️ On the circle

  $x^{2} +$$y^{2} > r^{2}$ ↔️ Outside the circle
• What is the radius of the circle?

  7
• What is the radius of the circle represented by the equation $(x + 1)^{2} +$$(y - 3)^{2} =$$49$?


  7
• How can you use the equation of a circle to find its center and radius?

  • The equation of a circle is (x-h)^2 + (y-k)^2 = r^2
  • (h, k) gives the coordinates of the center
  • r is the radius
• What is the equation of the example circle provided in the study material?

  x^{2} + y^{2} = 9</latex>
• At what rate does the area of the circle change in the related rates example?

  $2 \text{ cm}^{2} / \text{s}$
• Steps to graph a circle using its equation

  1️⃣ Identify the center $(a, b)$ and plot it

  2️⃣ Find the radius $r$ by taking the square root

  3️⃣ Draw the circle with the identified center and radius
• What is the equation of the circle graphed in the image?

  $(x+1)^2 +$$(y-3)^2 =$$49$
• What is the first step in graphing a circle using its equation?

  Identify and plot the center