A locus is a rule that draws a picture
Picture this: pin a point F on the floor and ask for every spot that sits exactly 3 metres from it. Walk it out and you trace a circle. That set — all the points obeying one condition, and no others — is a locus. The word just means 'place' in Latin, and the idea is the soul of this whole rung: a geometric shape is nothing but the collection of points that pass a test. In the earlier rungs a circle was a drawn curve; here it is reborn as the answer to a question about distance.
The miracle is that the test translates into algebra. 'Distance from F equals 3' becomes, through the distance formula you mastered in guide 2, the equation (x - a)^2 + (y - b)^2 = 9 — exactly the equation of a circle from guide 4. So a locus is a condition on points; an equation is that same condition written in coordinates; and the curve is the picture both describe. Three faces of one thing. Once you believe this, finding a curve is no longer drawing — it is reading a sentence about distance and copying it down with the distance formula.
Deriving a locus from scratch: the perpendicular bisector
Let us build one with our bare hands. Fix two points A and B and ask for every point P that is equidistant from them — same distance to A as to B. From pure geometry you already know the answer from an earlier rung: it is the perpendicular bisector of segment AB. The thrill of coordinate geometry is watching that classical fact fall out of algebra with no picture at all, just by writing the condition |PA| = |PB| and letting the symbols do the work.
- Place A = (1, 2) and B = (5, 4), and let P = (x, y) be a general point obeying the rule.
- Write |PA| = |PB| with the distance formula, then square both sides to kill the square roots: (x - 1)^2 + (y - 2)^2 = (x - 5)^2 + (y - 4)^2.
- Expand both sides. The x^2 and y^2 terms appear identically on the left and right, so they cancel — this cancellation is the whole point, and it is why the locus turns out to be a straight line.
- What remains is linear: -2x - 4y + 5 = -10x - 8y + 41, which tidies to 8x + 4y = 36, that is 2x + y = 9.
- Check it: the midpoint of AB is (3, 3), and 2(3) + 3 = 9 holds, so the midpoint lies on the line. The line's slope is -2, while AB has slope 1/2, and (-2)(1/2) = -1 — perpendicular, exactly as the name promised.
Look at what just happened: a condition about distance produced a line, and the line passed straight through the midpoint of AB at right angles. We did not assume any of that — the algebra forced it. This is the characteristic pleasure of the method: you state a rule, you grind the symbols, and a theorem you half-remembered from synthetic geometry emerges proven, not merely recalled.
Intersections: where two curves agree
If a curve is the set of points satisfying one equation, then where two curves cross is the set of points satisfying both at once. That is the entire idea of an intersection: solve the two equations as a simultaneous system, and each solution pair (x, y) is a crossing point. Two lines are the simplest case — a system of two lines — and you already know the three outcomes: one solution (they cross), no solution (parallel, never meeting), or infinitely many (the same line twice). Geometry and algebra agree perfectly, because a solution is a shared point.
The richer case is a line meeting a circle, and here algebra tells you something the eye cannot guarantee. Substitute the line into the circle's equation and you always land on a quadratic in one variable. A quadratic has at most two real roots, and that number is decided by its discriminant: two roots means the line is a secant cutting the circle twice; one repeated root means the line is a tangent, grazing at a single point; and no real root means the line misses the circle entirely. The discriminant is doing the geometry for you — it counts the intersections before you have drawn a thing.
Intersect the line y = x + 1 with the circle x^2 + y^2 = 25.
Substitute y = x + 1 :
x^2 + (x + 1)^2 = 25
2x^2 + 2x + 1 = 25
2x^2 + 2x - 24 = 0
x^2 + x - 12 = 0
(x + 4)(x - 3) = 0 -> x = -4 or x = 3
Back-substitute into y = x + 1 :
x = -4 -> P = (-4, -3)
x = 3 -> Q = ( 3, 4)
Two real roots == a SECANT, cutting the circle at P and Q.
(One repeated root would be a tangent; no real root, a clean miss.)Coordinate proof: geometry settled by algebra
Now the rung's payoff. A coordinate proof takes a theorem of plain geometry, drops the figure onto the coordinate plane, and settles it by computation instead of a chain of synthetic steps. The art is in the setup: place the figure so its coordinates are as clean as possible, because a smart choice of axes can turn a fearsome proof into one line of algebra. Crucially, the placement must stay general — you may use the plane's freedom to pin down position and orientation, but you must not secretly assume the very thing you are trying to prove.
Take a famous one: the midpoints of the four sides of any quadrilateral form a parallelogram (Varignon's theorem). Synthetically it needs the midsegment idea twice; in coordinates it is almost embarrassing. Put the four vertices at A = (0, 0), B = (2b, 0), C = (2c, 2d), D = (2e, 2f) — the 2's are a tidy trick so the midpoints come out without fractions. The side midpoints are P = (b, 0), Q = (b + c, d), R = (c + e, d + f), S = (e, f). Now compute the vector from P to Q and the vector from S to R: both equal (c, d). Opposite sides equal and parallel — that is a parallelogram. Done, and it held for every quadrilateral because we never assumed anything special about b, c, d, e, f.
What this rung has quietly given you
Step back and see the shape of the whole rung. A grid let geometry and algebra share one language; distance, midpoint, and slope became formulas; lines and circles became equations. This last guide closes the loop by letting equations do geometry back: a locus is a rule made into a curve, an intersection is a system you solve, and a coordinate proof settles a theorem by computation. Descartes' insight in the 1630s was exactly this — that the plane of geometry and the algebra of equations are two views of one world — and you have now used it in all three directions.
Be honest about the limits, though, because that honesty is what carries you up the ladder. Coordinate methods are powerful but not always the graceful route — a slick synthetic proof can be shorter and more illuminating than a page of algebra, and choosing the right tool is itself a skill. And the plane is only the beginning. The very same machinery — points as number-tuples, distance as a formula, shapes as equations — extends to three dimensions and beyond, where it grows into vectors, and eventually into the curved spaces and deep theorems that crown this entire ladder. You have just learned the dictionary; the rungs above show how much it can say.