Geometry & Trigonometry · Core Tested Skill (~15% of Math)
Circle Equations & Completing the Square
Key SAT Circle Rule at a Glance
The standard equation of a circle in the xy-plane is (x - h)² + (y - k)² = r², where (h, k) is the center point and r is the radius (always √r²). Pay close attention to signs: (x + 5)² + (y - 3)² = 49 has center (-5, 3) and radius 7.
1. The Anatomy of the Standard Circle Formula
Every circle question on the Digital SAT tests either finding the center and radius or relating lines (tangents or diameters) to the equation.
(x - h)² + (y - k)² = r²
Center is (h, k) with reversed signs from the parentheses. Radius is r = √r².
Watch Opposites
(x - 4)² implies h = +4. If it reads (x + 6)², then h = -6. Never forget that standard form subtracts coordinates.
The Right Side is r²
If the equation equals 36, the radius is 6, not 36. If it asks for diameter, double the radius (2r = 12).
Perpendicular to Radius
A tangent line touches the circle at one point and is perpendicular to the radius drawn to that point (m_tangent = -1 / m_radius).
When converting $x^2 + y^2 - 10x + 4y = 20$ into standard form, whatever square constants you add to the left side ($(-10/2)^2 = 25$ and $(4/2)^2 = 4$) must also be added to the right side: $20 + 25 + 4 = 49$. Forgetting to add them to the right is the #1 reason students pick incorrect distractor answers.
2. Fast Completing the Square in 3 Steps
When the College Board provides an expanded equation like $x^2 + y^2 + 12x - 8y - 12 = 0$:
- Group variables and move constants: $(x^2 + 12x) + (y^2 - 8y) = 12$
- Take half the linear coefficient and square it: Half of $12$ is $6$ ($6^2 = 36$). Half of $-8$ is $-4$ ($(-4)^2 = 16$).
- Add to both sides and collapse: $(x + 6)^2 + (y - 4)^2 = 12 + 36 + 16 = 64$. Center is $(-6, 4)$ and radius is $8$.
3. The Digital SAT Desmos Hack
On the Digital SAT, you do not even need to complete the square by hand if time is running low:
- Type the given equation directly into Desmos:
x^2 + y^2 - 8x + 6y = 56 - Desmos immediately renders the full circle!
- Click on the leftmost and rightmost edges of the circle to see the horizontal diameter endpoints. The center is the midpoint, and the distance from center to edge is your radius.
Worked Official-Style SAT Example
Question: In the $xy$-plane, the circle defined by $(x - 3)^2 + (y + 2)^2 = 25$ has diameter $AB$. If point $A$ has coordinates $(3, 3)$, what are the coordinates of point $B$?
View Step-by-Step Solution & Desmos Method
Correct Answer: A
Step 1: Identify Center: From the equation, the center is $(h, k) = (3, -2)$.
Step 2: Use Midpoint Logic: Since $AB$ is a diameter, the center $(3, -2)$ is the midpoint between $A(3, 3)$ and $B(x_B, y_B)$.
For the $x$-coordinate: $x_B = 3$.
For the $y$-coordinate: (3 + y_B) / 2 = -2 → 3 + y_B = -4 → y_B = -7.
Therefore, point $B$ is $(3, -7)$. Alternatively, in Desmos, graph the circle and the line $x = 3$; the two intersection points are $(3, 3)$ and $(3, -7)$!
Train this skill with active recall
Mindupload serves real circle geometry variants calibrated to the exact difficulty of Digital SAT Module 2.
Practice Circle Geometry on Mindupload