Analytic geometry of the plane: synthesis and challenges

Summary

This lesson closes the unit devoted to the analytic geometry of the plane. It gathers into a deductive chain what was proved in its nine lessons, from the distance formula to the vertex theorem of linear programming; it shows how the unit rests on the earlier ones, in particular on completing the square, the discriminant and sign charts, and how it prepares the following unit, whose transcendental functions describe the hyperbola once again. It then offers an exit assessment, parallel to the entry assessment, and ends with ten challenges, harder than the problems proposed in the lessons, which require combining several of them and finding in each case the idea that opens the way.

Learning objectives

State the main results of the unit and explain how each one depends on the earlier ones.
Relate the methods of the unit to those of the preceding units and to those of the following unit.
Solve problems that combine lines, vectors, conics and inequalities, choosing among the metric definition of a curve, its equation and the discriminant the most effective tool.
Translate conditions of minimum, of tangency and of locus into equations, and verify that the equation obtained describes exactly the set sought.

What the unit achieved

Coordinates, lines and vectors

The unit began with a single idea: identifying each point of the plane with the pair of its coordinates. It followed, in the lesson on the equation of a line, that the distance between two points is a consequence of the Pythagorean theorem and that the slope of a line does not depend on the points with which it is computed, by the similarity of triangles. These two theorems, together with the free choice of the axes, the equality of corresponding angles between parallels, the sum of the angles of a triangle and the angles of the isosceles triangle, are the facts of Euclidean geometry that the unit explicitly admits at the outset, because their proof belongs to the axiomatic construction of that geometry. From the invariance of the slope it was deduced that the points of a line are exactly the solutions of an equation $Ax + By + C = 0$ ; from a computation with the equations of two lines, that two non-vertical lines are parallel if and only if they have the same slope; from the Pythagorean theorem and its converse, together with the equality of corresponding angles, that they are perpendicular if and only if the product of their slopes is $-1$ ; and from the Pythagorean theorem, that the perpendicular dropped from a point is the shortest path to the line.

The lesson on vectors in the plane reformulated these results without depending on the choice of the axes. Vectors, with their addition and scalar multiplication, satisfy the eight vector space axioms; the dot product decides perpendicularity through the condition $\vec{u} \cdot \vec{v} = 0$ , which no longer excludes vertical lines, and yields the Cauchy-Schwarz inequality, the triangle inequality and the orthogonal projection. From this were obtained the vector equation of a line, the normal vector $\left( A, B \right)$ of the line $Ax + By + C = 0$ and the formula for the distance from a point to a line, which reappears in almost all the subsequent lessons:

\operatorname{d}(P, r) = \frac{\abs{A x_1 + B y_1 + C}}{\sqrt{A^2 + B^2}}

Conics defined by distances

The five lessons that came after applied one and the same method to four curves: define the curve by a property of distances, translate it into an equation by squaring, and then prove the converse, because squaring may add extraneous points. The lesson on the equation of parabolas defined the parabola by a focus and a directrix, proved that the graphs of quadratic polynomials are exactly the parabolas with a vertical axis and established their reflection property; the one on characterizing parabolas obtained, by completing the square, their elements in terms of the coefficients, and proved that the vertex is the extreme point and that the discriminant decides the intercepts with the horizontal axis.

The lesson on ellipses and circles defined the ellipse by the constant sum $2a$ of the distances to two foci, deduced its reduced equation with $b^2 = a^2 - c^2$ and proved that the distances from each point to the foci are $a + ex$ and $a - ex$ ; the circle turned out to be the ellipse with coincident foci. The lesson on the circle measured the position of a point by its power, decided that of a line by comparing the distance from the center with the radius, proved that the tangent is perpendicular to the radius, classified the six positions of two circles by means of their radical axis and proved the power of a point theorem, from which follow the intersecting chords theorem, the tangent-secant theorem and the radical center theorem. The lesson on hyperbolas replaced the sum by the difference, with $c^2 = a^2 + b^2$ , studied the asymptotes, proved that the graph of $y = \frac{1}{x}$ is a rectangular hyperbola and showed that a line parallel to an asymptote meets the hyperbola in a single point without being tangent.

The unified theory and inequalities

The lesson on conic sections brought the four curves together into a single theory. Every equation $Ax^2 + Cy^2 + Dx + Ey + F = 0$ represents, according to the sign of $AC$ and that of the right-hand side $G$ left by completing the squares, an ellipse, a hyperbola, a parabola or one of their degenerate cases; and the focus-directrix theorem describes the ellipse, the parabola and the hyperbola as the loci in which the ratio between the distance to a focus and the distance to a directrix is a fixed number $e$ , less than, equal to or greater than $1$ .

The last lesson, on inequalities in the plane and linear programming, passed from equalities to inequalities. The half-plane theorem ensures that the expression $Ax + By + C$ keeps its sign on each side of the line, whence the test point method; the regions defined by linear inequalities are convex, because a linear expression varies along a line segment between its values at the endpoints; a second-degree expression that does not vanish on a line segment keeps its sign on it, and that is why the test point method also works for regions bounded by conics; and a linear function attains its maximum and its minimum over a bounded convex polygon at one of its vertices. If the region is not bounded, the optimum may fail to exist, and even when it exists it may fail to be attained at a vertex, as in a strip; but if the region has some vertex, that is, if it contains no entire lines, every optimum that exists is also attained at one of them.

Three tools run through the whole unit. The first is the distance formula, on which all the definitions of curves rest. The second is completing the square, which brings each second-degree equation to the form in which its elements can be read. The third is the discriminant, which decides for all conics whether a line meets them in two points, in one or in none, and which, with the caveat about lines parallel to the axis of the parabola or to the asymptotes of the hyperbola, characterizes tangents without resorting to limits.

The following figure gathers the three segments into a single chain. Each node of the column is a definition or a theorem, numbered according to the logical order and joined by an arc to the earlier links that its proof uses; around it float the supports, that is, the principles and tools to which the chain resorts again and again, each one joined by a solid line to the links that rest on it and by a dotted one to the link where it is established, except those that are admitted or come from earlier units, which are drawn as rings. Clicking a support highlights all the links that use it, and each link refers to the lesson in which it is proved.

Supports of the chain

AEuclidean geometry. Admitted at the beginning of the unit are the free choice of the axes, the Pythagorean theorem and its converse, the equality of corresponding angles between parallels, the intercept theorem and the similarity of triangles, the sum of the angles of a triangle and the equality of the angles opposite the equal sides of an isosceles triangle. Admitted without proof in this unit.
BDistance formula. The distance between $(x_1, y_1)$ and $(x_2, y_2)$ is $\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$ ; all the definitions of curves in the unit rest on it. Established in: 1. Distance and midpoint.
CGeneral equation of a line. The points of every line are exactly the solutions of an equation $Ax + By + C = 0$ , with $A$ and $B$ not both zero. Established in: 2. Equation of a line.
DDistance from a point to a line. The distance from the point $(x_1, y_1)$ to the line $Ax + By + C = 0$ is $\frac{\abs{A x_1 + B y_1 + C}}{\sqrt{A^2 + B^2}}$ . Established in: 10. Distance from a point to a line.
ECompleting the square. The identity $ax^2 + bx + c = a\left(x + \frac{b}{2a}\right)^2 - \frac{\Delta}{4a}$ holds, with $\Delta = b^2 - 4ac$ ; proved in the unit on algebraic functions. Admitted without proof in this unit.
FDiscriminant. A quadratic equation has two roots, a double one or none according as its discriminant is positive, zero or negative, and without roots the polynomial keeps the sign of its leading coefficient; proved in the unit on algebraic functions. Admitted without proof in this unit.
GTranslation of graphs. Replacing $x$ by $x - h$ and $y$ by $y - k$ translates a graph $h$ units horizontally and $k$ vertically; proved in the unit on algebraic functions. Admitted without proof in this unit.
HRule of signs. A product of two factors is positive if they have the same sign and negative if they have opposite signs; it is the foundation of the sign charts of the unit on algebraic functions. Admitted without proof in this unit.

Segment 1 Coordinates, lines and vectors
- Obtain from the Pythagorean theorem the distance between two points and the midpoint of a line segment.
- Prove that the points of a line are exactly the solutions of a first-degree equation, and characterize parallelism and perpendicularity by the slopes.
- Equip the vectors of the plane with addition, scalar multiplication, the norm and the dot product, and deduce the Cauchy-Schwarz inequality, the triangle inequality and the orthogonal projection.
- Obtain the vector equation of a line, its normal vector and the distance from a point to a line.
1. Distance and midpoint. The distance between $P_1 = (x_1, y_1)$ and $P_2 = (x_2, y_2)$ is $\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$ , and the point $\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$ divides the line segment into two equal halves.
  Rests on: A (Euclidean geometry). Proved in: The equation of a line and Cartesian coordinate systems.
2. Equation of a line. The slope does not depend on the points with which it is computed, and a point lies on the line of slope $m$ through $(x_0, y_0)$ if and only if $y - y_0 = m(x - x_0)$ ; hence every line has an equation $Ax + By + C = 0$ .
  Rests on: A (Euclidean geometry). Proved in: The equation of a line and Cartesian coordinate systems.
3. Parallel and perpendicular lines. Two distinct non-vertical lines are parallel if and only if they have the same slope, and perpendicular if and only if the product of their slopes is $-1$ .
  Uses: 2. Equation of a line. Rests on: A (Euclidean geometry), B (Distance formula). Proved in: The equation of a line and Cartesian coordinate systems.
4. The perpendicular is the shortest path. If $Q$ is the foot of the perpendicular dropped from $P$ to the line $r$ , every other point $R$ of $r$ satisfies $d(P, R) > d(P, Q)$ .
  Uses: 3. Parallel and perpendicular lines. Rests on: A (Euclidean geometry), B (Distance formula). Proved in: The equation of a line and Cartesian coordinate systems.
5. Operations on vectors. Vector addition and scalar multiplication satisfy the eight vector space axioms, and two nonzero vectors with $u_1 v_2 - u_2 v_1 \neq 0$ generate each vector of the plane in a unique way.
  Proved in: Vectors in the plane and parametric equations of a line.
6. Dot product and perpendicularity. The dot product $\vec{u} \cdot \vec{v} = u_1 v_1 + u_2 v_2$ is commutative and distributive, satisfies $\vec{u} \cdot \vec{u} = {\norma{\vec{u}}}^2$ , and two vectors are perpendicular if and only if $\vec{u} \cdot \vec{v} = 0$ .
  Uses: 5. Operations on vectors. Rests on: A (Euclidean geometry), B (Distance formula). Proved in: Vectors in the plane and parametric equations of a line.
7. Cauchy-Schwarz and the triangle inequality. One has $\abs{\vec{u} \cdot \vec{v}} \le \norma{\vec{u}} \norma{\vec{v}}$ , whence $\norma{\vec{u} + \vec{v}} \le \norma{\vec{u}} + \norma{\vec{v}}$ and, for three points, $d(P, R) \le d(P, Q) + d(Q, R)$ .
  Uses: 6. Dot product and perpendicularity. Rests on: F (Discriminant). Proved in: Vectors in the plane and parametric equations of a line.
8. Orthogonal projection. If $\vec{v}$ is not zero, there is a unique number $t$ such that $\vec{u} - t \vec{v}$ is perpendicular to $\vec{v}$ , namely $t = \frac{\vec{u} \cdot \vec{v}}{{\norma{\vec{v}}}^2}$ .
  Uses: 6. Dot product and perpendicularity. Proved in: Vectors in the plane and parametric equations of a line.
9. Vector equation and normal vector. The points of the line through $P_0$ with direction vector $\vec{d}$ are the points $P_0 + t \vec{d}$ , each with a unique parameter $t$ ; and $\vec{n} = \left( A, B \right)$ is perpendicular to the line $Ax + By + C = 0$ , whose points are the points $P$ with $\vec{n} \cdot \vec{P_0 P} = 0$ .
  Uses: 2. Equation of a line, 6. Dot product and perpendicularity. Rests on: C (General equation of a line). Proved in: Vectors in the plane and parametric equations of a line.
10. Distance from a point to a line. The distance from $(x_1, y_1)$ to the line $Ax + By + C = 0$ is $\frac{\abs{A x_1 + B y_1 + C}}{\sqrt{A^2 + B^2}}$ , and it is attained at the foot of the perpendicular.
  Uses: 4. The perpendicular is the shortest path, 8. Orthogonal projection, 9. Vector equation and normal vector. Rests on: C (General equation of a line). Proved in: Vectors in the plane and parametric equations of a line.
Segment 2 Conics defined by distances
- Define the parabola by a focus and a directrix, obtain its equation and prove that the graphs of quadratic polynomials are exactly the parabolas with a vertical axis.
- Read off, by completing the square, the vertex of the parabola and decide with the discriminant its intercepts with the horizontal axis.
- Deduce the reduced equation of the ellipse and the fundamental equation of the hyperbola, with the proof of the converse, and recognize the circle as the ellipse with coincident foci.
- Study the position of a line and of another circle with respect to a circle, and prove the power of a point theorem.
- Characterize the tangents to the conics by means of the discriminant, with the caveat about lines parallel to the axis or to the asymptotes, and treat vertical lines separately.
1. Equation of the parabola. The points equidistant from the focus $(h, k + f)$ and from the directrix $y = k - f$ are exactly those of the curve $(x - h)^2 = 4f(y - k)$ , symmetric about its axis $x = h$ .
  Rests on: B (Distance formula), G (Translation of graphs). Proved in: The equation of the parabola: definitions and properties.
2. Parabola and quadratic polynomial. The graphs of $y = ax^2 + bx + c$ , with $a \neq 0$ , are exactly the parabolas with a vertical axis, with $f = \frac{1}{4a}$ , whose absolute value is the focal distance.
  Uses: 11. Equation of the parabola. Rests on: E (Completing the square). Proved in: The equation of the parabola: definitions and properties.
3. Reflection property. The tangent at a point $P$ of the parabola makes equal angles with the parallel to the axis through $P$ and with the line joining $P$ to the focus.
  Uses: 11. Equation of the parabola. Rests on: A (Euclidean geometry), B (Distance formula), F (Discriminant). Proved in: The equation of the parabola: definitions and properties.
4. Vertex and intercepts with the axis. The vertex of $y = ax^2 + bx + c$ is $\left(-\frac{b}{2a}, -\frac{\Delta}{4a}\right)$ , its lowest point if $a > 0$ and its highest if $a < 0$ , and the parabola meets the $x$ -axis in two points, in one or in none according as $\Delta$ is positive, zero or negative.
  Uses: 12. Parabola and quadratic polynomial. Rests on: E (Completing the square), F (Discriminant). Proved in: Characterizing parabolas and their graphs.
5. Reduced equation of the ellipse. With $b^2 = a^2 - c^2$ , the ellipse with foci $(\pm c, 0)$ and sum of distances $2a$ is exactly the curve $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ , and the distances from each point to the foci are $a + \frac{c}{a} x$ and $a - \frac{c}{a} x$ .
  Rests on: B (Distance formula). Proved in: The equation of the ellipse and of the circle.
6. Equation of the circle. The circle with center $(h, k)$ and radius $r$ is the curve $(x - h)^2 + (y - k)^2 = r^2$ , and it is the ellipse whose two foci coincide with the center.
  Uses: 15. Reduced equation of the ellipse. Rests on: B (Distance formula), G (Translation of graphs). Proved in: The equation of the ellipse and of the circle.
7. Tangency to the ellipse. A non-vertical line is tangent to the ellipse if and only if the discriminant of the equation that results from substituting it is zero; if it is positive, the line meets the ellipse in two points, and if it is negative, it does not touch it. The only vertical tangents are $x = a$ and $x = -a$ .
  Uses: 15. Reduced equation of the ellipse. Rests on: F (Discriminant). Proved in: The equation of the ellipse and of the circle.
8. Position of a line. If $\delta$ is the distance from the center to a line, the line does not touch the circle if $\delta > r$ , touches it only at the foot of the perpendicular if $\delta = r$ and meets it in two points if $\delta < r$ .
  Uses: 4. The perpendicular is the shortest path, 16. Equation of the circle. Rests on: A (Euclidean geometry), D (Distance from a point to a line), F (Discriminant). Proved in: The circle: tangents, relative positions and power of a point.
9. The tangent is perpendicular to the radius. A line through a point $T$ of the circle is tangent if and only if it is perpendicular to the radius joining the center to $T$ ; through each point there passes a single tangent.
  Uses: 3. Parallel and perpendicular lines, 4. The perpendicular is the shortest path, 18. Position of a line. Proved in: The circle: tangents, relative positions and power of a point.
10. Radical axis and relative positions. The points of equal power with respect to two circles with distinct centers form a line perpendicular to the line of the centers, and the two circles have two common points if $\abs{r_1 - r_2} < d < r_1 + r_2$ , one if $d$ equals one of those bounds, in which case, apart from the point of contact, the points of the smaller lie inside the larger in internal tangency and those of each lie outside the other in external tangency, and none otherwise.
  Uses: 3. Parallel and perpendicular lines, 7. Cauchy-Schwarz and the triangle inequality, 16. Equation of the circle, 18. Position of a line. Rests on: C (General equation of a line), D (Distance from a point to a line). Proved in: The circle: tangents, relative positions and power of a point.
11. Power of a point. If a line through $P$ meets the circle at $A$ and $B$ , then $d(P, A) \cdot d(P, B) = \abs{p}$ , with $p = d^2 - r^2$ ; and from an exterior point there issue exactly two tangents, both of length $\sqrt{p}$ .
  Uses: 18. Position of a line, 19. The tangent is perpendicular to the radius, 20. Radical axis and relative positions. Rests on: A (Euclidean geometry). Proved in: The circle: tangents, relative positions and power of a point.
12. Equation of the hyperbola. With $b^2 = c^2 - a^2$ , the hyperbola with foci $(\pm c, 0)$ and difference of distances $2a$ is exactly the curve $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ , whose branches approach the asymptotes $y = \pm \frac{b}{a} x$ .
  Rests on: B (Distance formula). Proved in: The equation of the hyperbola and its derivation.
13. Rectangular hyperbola. The graph of $y = \frac{1}{x}$ is the hyperbola with foci $(\sqrt{2}, \sqrt{2})$ and $(-\sqrt{2}, -\sqrt{2})$ whose distances to them differ by $2\sqrt{2}$ ; it is rectangular and its asymptotes are the axes.
  Uses: 22. Equation of the hyperbola. Rests on: A (Euclidean geometry), B (Distance formula). Proved in: The equation of the hyperbola and its derivation.
14. Tangency to the hyperbola. A line not parallel to the asymptotes is tangent to the hyperbola exactly when the discriminant is zero; a line parallel to an asymptote, without coinciding with it, meets the hyperbola in a single point and crosses it; and the only vertical tangents are $x = a$ and $x = -a$ .
  Uses: 22. Equation of the hyperbola. Rests on: F (Discriminant). Proved in: The equation of the hyperbola and its derivation.
Segment 3 The unified theory and inequalities
- Classify the equations $Ax^2 + Cy^2 + Dx + Ey + F = 0$ by the sign of $AC$ and by the right-hand side left by completing the squares.
- Describe the ellipse, the parabola and the hyperbola as loci of a focus and a directrix, according to their eccentricity.
- Prove that a linear expression keeps its sign on each side of its line and base the test point method on this.
- Prove that the regions defined by linear inequalities are convex, extend the test point method to conics and prove that a linear function attains its extreme values over a bounded convex polygon at a vertex, and at a vertex also over an unbounded region with vertices when those values exist.
1. General second-degree equation. Every equation $Ax^2 + Cy^2 + Dx + Ey + F = 0$ , with $A$ and $C$ not both zero, represents an ellipse, a hyperbola or a parabola according as $AC$ is positive, negative or zero, or one of their degenerate cases, among which the right-hand side $G$ left by completing the squares decides.
  Uses: 11. Equation of the parabola, 15. Reduced equation of the ellipse, 16. Equation of the circle, 22. Equation of the hyperbola. Rests on: E (Completing the square), F (Discriminant), G (Translation of graphs). Proved in: Conic sections: characterization and graphs of the three conics.
2. Focus-directrix theorem. The points whose distance to a focus $F$ is $e$ times their distance to a directrix $L$ form an ellipse if $e < 1$ , a parabola if $e = 1$ and a hyperbola if $e > 1$ .
  Uses: 11. Equation of the parabola, 15. Reduced equation of the ellipse, 22. Equation of the hyperbola, 25. General second-degree equation. Rests on: A (Euclidean geometry), B (Distance formula), E (Completing the square). Proved in: Conic sections: characterization and graphs of the three conics.
3. Directrices of the ellipse and of the hyperbola. In the ellipse and in the hyperbola of eccentricity $e = \frac{c}{a}$ , the distance from each point to the focus $(c, 0)$ is $e$ times its distance to the line $x = \frac{a}{e}$ ; the circle, by contrast, has no directrix.
  Uses: 15. Reduced equation of the ellipse, 22. Equation of the hyperbola, 26. Focus-directrix theorem. Proved in: Conic sections: characterization and graphs of the three conics.
4. Half-plane theorem. The expression $Ax + By + C$ is positive at all the points of one of the sides of the line $Ax + By + C = 0$ and negative at all those of the other; hence the test point method.
  Rests on: C (General equation of a line), H (Rule of signs). Proved in: Inequalities in the plane and linear programming.
5. Expressions along a line segment. On the line segment with endpoints $P$ and $Q$ , a linear expression satisfies $L(P_t) = (1 - t) L(P) + t L(Q)$ , so that the solution regions of systems of linear inequalities are convex; and a second-degree expression that does not vanish on a line segment keeps its sign on it, so that each conic divides the plane into regions, two or three, in which the test point method holds.
  Uses: 2. Equation of a line, 28. Half-plane theorem, 7. Cauchy-Schwarz and the triangle inequality, 14. Vertex and intercepts with the axis. Rests on: F (Discriminant), H (Rule of signs). Proved in: Inequalities in the plane and linear programming.
6. Vertex theorem. A function $ax + by + c$ attains its maximum and its minimum over the nonempty, bounded region of a system of non-strict linear inequalities, each at one of its vertices; if the region is not bounded but has a vertex, that is, if it contains no entire lines, every maximum or minimum that is attained is also attained at a vertex.
  Uses: 29. Expressions along a line segment. Proved in: Inequalities in the plane and linear programming.

Towards the transcendental functions. The chain culminates the study of algebraic curves of degree one and two. The following unit leaves the algebraic functions, but the hyperbola reappears in it: the areas under $y = \frac{1}{x}$ behave like logarithms, and the point $(\cosh t, \sinh t)$ traces the right branch of $x^2 - y^2 = 1$ .