How Curves and Equations Correspond

Try this

“The equation of the curve $C$ is $F (x, y) = 0$ ” demands that two things both be true:

Purity: the points of the curve all satisfy the equation. Put the other way round, every point given by a solution of the equation satisfies the condition: no extra points have slipped in;
Completeness: every point that satisfies the condition is on the curve: not a single one is missed.

In the widget, the small dots are the locus points the mechanism “really” produces, and the dashed curve is an equation somebody derived. $Q$ is your test point, and two readouts check “does it satisfy the condition?” and “does it satisfy the equation?” separately.

Choose “Chord midpoints” and drag the slider from $- 35$ to $35$ . Which part of the dashed circle do the dots cover? Beyond what $∣ φ ∣$ does the line stop cutting the circle?
Press “ $Q \to (2, 2)$ ” and “ $Q \to P_{0}$ itself”. What are the verdicts?
Choose “Right-angle vertex” and drag the slider back and forth. What happens at $φ = 0$ ? Then send $Q$ to $A$ .
Choose “Equidistant from the axes” and drag the slider through its range. How many lines do the dots draw? How many dashed lines are there? Send $Q$ to $(2, - 2)$ .
Of the four verdicts (genuine, extra, missing, unrelated), which one shows the equation is not pure? Which one shows it is not complete?

Predict first

Lines through $P_{0} (4, 0)$ cut chords from the circle $x^{2} + y^{2} = 4$ , and the chord midpoints lie on the circle $(x - 2)^{2} + y^{2} = 4$ . How much of that circle do they occupy?

The whole circleHalf the circleOne third of the circleThe whole circle except

P_{0}

Choose an answer first; then you can see the result.

Why

The standard procedure for a locus problem is: set up coordinates and a moving point → write the condition → turn it into an equation → check. The last step is exactly the check of purity and completeness. Here is a complete run.

Example: lines through P₀(4, 0) cut the circle x² + y² = 4; find the locus of the chord midpoint M

Step 1 / 5

Find the condition
The line joining the midpoint of a chord to the center is perpendicular to the chord: $O M ⊥ M P_{0}$ . So $∠ O M P_{0} = 9 0^{\circ}$ , and $M$ lies on the circle with diameter $O P_{0}$ .

$O M \cdot P_{0} M = 0$
Turn it into an equation
Is it complete?
Is it pure?
Remove the extra part

Notice that the final range was dug out of the condition, not out of the equation: the equation itself knows nothing about the given circle. That is exactly why purity has to be checked separately.

Practice

Exercise 1Basic

$A (0, 0)$ , $B (4, 0)$ . The locus of points $P$ with $P A ⊥ P B$ is the circle $(x - 2)^{2} + y^{2} = 4$ with some points removed. How many points are removed?

Exercise 2Basic

How many lines make up the locus of points whose distance to the $x$ -axis is $2$ times their distance to the $y$ -axis?

Exercise 3Advanced

Chords of the circle $x^{2} + y^{2} = 9$ are drawn through the interior point $P_{0} (1, 0)$ . The locus of their midpoints is a complete circle. What is its radius?

Exercise 4Challenge

Lines through $P_{0} (2, 0)$ cut chords from the circle $x^{2} + y^{2} = 1$ . The locus of the chord midpoints is the arc of $(x - 1)^{2} + y^{2} = 1$ with $0 \leq x < k$ . Find $k$ .

Try this

Why

Example: lines through P₀(4, 0) cut the circle x² + y² = 4; find the locus of the chord midpoint M

Practice

Mistake Museum · Exhibits from this stop

Removing an absolute value and keeping only one case

Counting A and B in the locus of a right-angle vertex