Day 6 — Coordinate Geometry
Time: ~30 min · Topic: coordinate geometry
The problem
A circle has its center at (3, -2) and passes through the point (7, 1). What is the equation of the circle?
Hint: Find the radius first — it's just the distance from the center to any point on the circle.
Try it yourself
Take a few minutes. Don't peek at the steps. If you get stuck, the hint above is enough to nudge you forward.
Step-by-step solution
- Use the distance formula to find the radius: r = sqrt((7 - 3)^2 + (1 - (-2))^2) = sqrt(16 + 9) = sqrt(25) = 5
- Recall the standard form of a circle: (x - h)^2 + (y - k)^2 = r^2, where (h, k) is the center.
- Plug in center (3, -2) and r = 5: (x - 3)^2 + (y + 2)^2 = 25
Answer: (x - 3)^2 + (y + 2)^2 = 25
Why this matters
The equation of a circle comes directly from the distance formula — which was made possible by Rene Descartes inventing the coordinate plane in 1637, literally the same system you're using to solve this problem!
This day in math: The Man Who Tamed Randomness
April 26, 1903 · Andrei Kolmogorov
On April 25, 1903, Andrei Kolmogorov was born in Tambov, Russia. He would go on to become one of the most influential mathematicians of the 20th century, revolutionizing probability theory with his 1933 monograph that laid down just three elegant axioms — giving the entire field its rigorous mathematical foundation for the first time. His work spanned an astonishing range: turbulence theory, algorithmic complexity (now called Kolmogorov complexity), topology, and classical mechanics.
Why it matters: Before Kolmogorov, probability was more intuition than mathematics. His axioms unified the field under measure theory, making modern statistics, machine learning, quantum mechanics, and financial modeling possible.
Did you know?
If you randomly pick two whole numbers, the probability that they share no common factor (other than 1) is exactly 6 / pi^2 — roughly 61%. Pi, a number born from circles, mysteriously appears in a problem that has absolutely nothing to do with geometry.
This result comes from the Euler product over all prime numbers. The chance that two random numbers are both NOT divisible by a given prime p is (1 - 1/p^2). Multiplying this across every prime gives 1/zeta(2), and the great mathematician Euler proved that zeta(2) = pi^2 / 6. So the final probability simplifies beautifully to 6 / pi^2, connecting prime numbers to circles in one of math's most unexpected bridges.
Done?
- Solved the problem (or read through the steps)
- Read the history note
- Read the "did you know" fact