Identifying Center and Radius

Find the center and radius of the circle: $(x - 3)^2 + (y + 2)^2 = 25$

Compare to standard form: Standard form: $(x - h)^2 + (y - k)^2 = r^2$ = Match the pattern

Identify h (x-coordinate of center): $(x - 3)^2$ means $h = 3$ = $h = 3$

Identify k (y-coordinate of center): $(y + 2)^2 = (y - (-2))^2$ means $k = -2$ = $k = -2$

Find the radius: $r^2 = 25$, so $r = \sqrt{25} = 5$ = $r = 5$

Answer: Center: $(3, -2)$, Radius: $5$

Writing an Equation from Center and Radius

Write the equation of a circle with center $(-4, 1)$ and radius $6$.

Identify the values: $h = -4$, $k = 1$, $r = 6$ = Values identified

Substitute into standard form: $(x - (-4))^2 + (y - 1)^2 = 6^2$ = Substitute values

Simplify: $(x + 4)^2 + (y - 1)^2 = 36$ = Final equation

Answer: $(x + 4)^2 + (y - 1)^2 = 36$

Circle Through a Point

Write the equation of a circle centered at $(2, -1)$ that passes through the point $(5, 3)$.

Find the radius using distance formula: $r = \sqrt{(5-2)^2 + (3-(-1))^2}$ = $r = \sqrt{9 + 16}$

Calculate the radius: $r = \sqrt{25} = 5$ = $r = 5$

Write the equation: $(x - 2)^2 + (y + 1)^2 = 25$ = Standard form equation

Answer: $(x - 2)^2 + (y + 1)^2 = 25$

Mistake: Confusing the signs in $(y + 2)^2$

Why: Standard form uses subtraction: $(y - k)^2$. If you see $(y + 2)^2$, this equals $(y - (-2))^2$, so $k = -2$, not $+2$.

Correct: Always rewrite as subtraction: $(y + 2)^2 = (y - (-2))^2$, therefore $k = -2$.

Mistake: Using $r^2$ as the radius instead of $r$

Why: The equation gives $r^2$, not $r$. You must take the square root.

Correct: If the equation shows $= 49$, then $r^2 = 49$ and $r = 7$ (not 49).

Mistake: Forgetting to square the radius when writing equations

Why: When given radius $r = 4$, students write $= 4$ instead of $= 16$.

Correct: Always square the radius: if $r = 4$, write $r^2 = 16$ in the equation.

GPS Triangulation

Cell towers and GPS satellites use circles to locate your position. Each tower knows its distance to your phone, creating a circle of possible locations.

A cell tower at position $(2, 3)$ km detects your phone at distance $5$ km. Your possible locations form the circle $(x - 2)^2 + (y - 3)^2 = 25$.

Circular Race Tracks

Architects design circular tracks using circle equations to ensure the track has the correct dimensions.

A running track has an inner edge with center at the origin and radius $30$ meters: $x^2 + y^2 = 900$. The outer edge has radius $35$ meters: $x^2 + y^2 = 1225$.

Q: Why is there a minus sign in $(x - h)$?

A: The standard form uses subtraction because it's based on the distance formula. The expression $(x - h)^2 + (y - k)^2$ calculates the squared distance from any point $(x, y)$ to the center $(h, k)$.

Q: Can the radius be negative?

A: No. The radius is always positive because it represents a distance. If $r^2 = 25$, then $r = 5$ (we take the positive square root).

Q: What if the equation doesn't look like standard form?

A: Some equations need to be rewritten. For example, $x^2 + y^2 - 6x + 4y = 12$ can be converted to standard form by completing the square. This is covered in the 'Converting to Standard Form' lesson.