Skip to content

Analytic geometry, grade 12 (5 units)

Geometric shapes as equations in the coordinate system: the line, the circle, the parabola, the ellipse and the hyperbola.Subjects

The topics, in order:

  • points and lines: dividing a segment in a given ratio, the distance of a point from a line and between parallel lines, the positions of two lines, loci that are lines;
  • the circle: the circle as a locus, completing the square, the tangent to a circle, a line and a circle, two circles, loci that are circles;
  • the parabola: the focus and the directrix, the canonical equation, a parabola with a line and with a circle, the tangent to a parabola;
  • the ellipse: two definitions, the canonical equation, an ellipse with a line and with a circle;
  • the hyperbola: the canonical equation and the asymptotes, a hyperbola with a line and with a circle;
  • all the curves together, and finding loci.

The course goes on from grade 10’s introduction to analytic geometry (5 units): the distance between two points, the midpoint of a segment, the equation of a line, parallel and perpendicular lines, and a circle with its centre at the origin.

There are two kinds of lessons:

  • learning lessons – an explanation, worked examples and a few questions to try;
  • practice lessons – questions only.

How the page works: Using a page.

  1. Recap: points, segments and linesLearningWhat grade 10 taught: the distance between two points, the midpoint of a segment, the slope and the equation of a line in explicit and implicit form, parallel and perpendicular lines, and where two lines meet.
  2. Dividing a segment in a given ratioLearningThe point dividing a segment internally in the ratio $k:l$: the formula, where it comes from, finding an end of a segment, and where the medians of a triangle meet.
  3. The distance of a point from a line; the positions of two linesLearningThe distance between two geometric objects, the formula for the distance of a point from a line, the distance between parallel lines, and lines that cross, are parallel or coincide.
  4. Loci that are linesLearningA locus and how to find it in the coordinate system: the perpendicular bisector, the points at a given distance from a line, the bisectors of the angles between two lines, and a vertex of a triangle of given area.
  5. Practice: points and linesPracticeDividing a segment, the medians' meeting point, the distance of a point from a line and between parallel lines, a triangle's area and loci that are lines.
  6. The circle's equation and completing the squareLearningThe circle as a locus, the canonical and the general equation, the expanded equation $x^{2}+y^{2}+Cx+Dy+E=0$, finding the centre and the radius by completing the square, and when such an equation is a circle.
  7. The tangent to a circle at a point on itLearningThe tangent is perpendicular to the radius at the point of contact: the tangent's equation by slopes and by the formula, a tangent parallel to an axis, a tangent that moves with the circle, and tangents of a given slope.
  8. A line and a circle: their positionsLearningA line cuts a circle, touches it or misses it: by the centre's distance from the line and by the discriminant of the system; the length of a chord, and a parameter that decides the position.
  9. Two circles: their positionsLearningTwo circles apart, touching, crossing or one inside the other – by the distance between the centres; the system of two circles' equations, and the common chord that subtracting the equations gives.
  10. Loci that are circlesLearningLoci whose equation is a circle: a fixed ratio of the distances from two points, a right angle standing on a segment, and the midpoints of the segments from a fixed point to a circle.
  11. Practice: the circlePracticeCompleting the square, the tangent's equation, a line and a circle, two circles, a chord and a locus that is a circle.
  12. The parabola: a locus and its canonical equationLearningThe parabola as the locus of the points as far from a focus as from a directrix; the canonical equation $y^{2}=2px$, the focus, the directrix and the symmetry; a shifted parabola.
  13. A parabola with a line and a circle; the tangent to a parabolaLearningThe positions of a line and a parabola, and of a circle and a parabola, by a system of equations, a line parallel to the axis of symmetry, and the tangent's equation $y\cdot y_0=p(x+x_0)$.
  14. The ellipse: two definitions and the canonical equationLearningThe ellipse as the locus of the points whose distances from two foci add up to a constant, and as a squashed circle; the canonical equation, the axes, the vertices and the foci.
  15. An ellipse with a line and a circleLearningThe positions of a line and an ellipse by a system of equations and the discriminant, a line touching an ellipse, and a circle with its centre on an axis with an ellipse.
  16. Practice: the parabola and the ellipsePracticeThe equations of a parabola and an ellipse, the tangent to a parabola, a line and a circle with a parabola and with an ellipse, and distances from the foci.
  17. The hyperbola: a locus, its equation and asymptotesLearningThe hyperbola as the locus of the points whose distances from two foci differ by a constant; the canonical equation, the vertices, the two branches, the foci and the asymptotes.
  18. A hyperbola with a line and a circleLearningThe positions of a line and a hyperbola – a line parallel to an asymptote, a line that misses it and a line that touches it – and a circle with its centre on an axis with a hyperbola.
  19. The curves together; finding a locusLearningThe equation $Ax^{2}+By^{2}+Cx+Dy+E=0$ – a circle, an ellipse, a hyperbola or a parabola, by its coefficients; shifted curves; and loci that are a parabola, an ellipse or a hyperbola.
  20. Practice: the hyperbola and lociPracticeThe equation of a hyperbola, a line and a circle with a hyperbola, a tangent by the discriminant, recognising a shifted curve, and loci.
  21. Review: analytic geometryPracticeQuestions from the whole course: dividing a segment and the distance of a point from a line, a circle and its tangent, two circles, a parabola and a line, an ellipse, a hyperbola and a locus.
Last updated on