Core Formula 01
Distance Formula
The distance between two points (x₁, y₁) and (x₂, y₂) in the coordinate plane is derived from the Pythagorean theorem:
d = √[(x₂ − x₁)² + (y₂ − y₁)²]
Key insight: You are finding the hypotenuse of a right triangle whose legs have lengths |x₂ − x₁| and |y₂ − y₁|.
- Always square the differences (negatives disappear).
- Order doesn't matter: (x₂ − x₁)² = (x₁ − x₂)².
- The result is always ≥ 0.
Core Formula 02
Midpoint Formula
The midpoint M between (x₁, y₁) and (x₂, y₂) is:
M = ( (x₁ + x₂)/2 , (y₁ + y₂)/2 )
Think: average the x-coordinates, average the y-coordinates.
Core Formula 03
Standard Form of a Circle
A circle with center (h, k) and radius r satisfies:
(x − h)² + (y − k)² = r²
- Center: (h, k) — watch the signs! If the equation is (x + 3)², the center's x-coordinate is −3.
- Radius: r = √(right-hand side). Make sure to take the square root of r².
- Circle centered at origin: (h=0, k=0) → x² + y² = r²
Core Formula 04
General Form → Standard Form
The general form is: x² + y² + Dx + Ey + F = 0
To convert to standard form, complete the square for both x and y:
Step 1: Group x-terms and y-terms
(x² + Dx) + (y² + Ey) = −F
Step 2: Add (D/2)² and (E/2)² to both sides
Step 3: Write as (x − h)² + (y − k)² = r²
Critical rule: Whatever you add to the left side, you must also add to the right side.
★ Key Facts to Memorize
01
Distance formula: d = √[(x₂−x₁)² + (y₂−y₁)²] — always add the squared differences.
02
Standard circle: (x−h)² + (y−k)² = r² — center (h, k), radius r (not r²!).
03
Sign flip: (x + 3)² means center x = −3. The equation is (x − (−3))².
04
Radius vs. r²: if the right side is 25, then r = 5 (not 25). Area = πr² = 25π.
05
Completing the square: add (b/2)² to both sides for x² + bx.
06
Midpoint = average of coordinates: M = ((x₁+x₂)/2, (y₁+y₂)/2).
07
A point is ON the circle if substituting it gives exactly r². Inside if < r². Outside if > r².
Worked Examples
Example 1 — Distance Formula
Find the distance between A(1, 2) and B(4, 6).
d = √[(4−1)² + (6−2)²] = √[3² + 4²] = √[9 + 16] = √25 = 5
Answer: d = 5
Example 2 — Identify Center and Radius
Find the center and radius of (x − 3)² + (y + 2)² = 49.
Rewrite: (x − 3)² + (y − (−2))² = 7²
Center = (3, −2), r = √49 = 7
Answer: Center (3, −2), Radius = 7
Example 3 — Write Equation from Center and Radius
Write the equation of the circle with center (−1, 4) and radius 5.
(x − (−1))² + (y − 4)² = 5²
(x + 1)² + (y − 4)² = 25
Example 4 — Completing the Square
Convert x² + y² − 6x + 4y − 12 = 0 to standard form.
(x² − 6x) + (y² + 4y) = 12
(x² − 6x + 9) + (y² + 4y + 4) = 12 + 9 + 4
(x − 3)² + (y + 2)² = 25
Center (3, −2), Radius = 5
Example 5 — Is a Point on the Circle?
Is (7, −2) on the circle (x − 3)² + (y + 2)² = 16?
Substitute: (7−3)² + (−2+2)² = 4² + 0² = 16 + 0 = 16 ✓
Yes — the point lies ON the circle.