151 Analytic Geometry Terms and Definitions | Mathematics Board Exam Review

INTRODUCTION

Analytic geometry is the bridge between algebra and geometry; the subject that lets you describe shapes, curves, and spatial relationships using equations and coordinates. For Philippine engineering licensure examinees, it is one of the most consistently tested mathematics topics across all engineering programs. The PRC board exam does not just ask you to identify formulas. It asks you to apply them quickly and correctly under pressure, which means your understanding of the vocabulary has to be both deep and fast.

This expanded reference covers 151 terms drawn from the complete scope of Analytic Geometry as tested in the PRC examinations. The original 101-term list has been fully revised and expanded to include three-dimensional analytic geometry, parametric equations, rotation of axes, polar forms of conics, radical axis and circle systems, and a more complete treatment of curve properties and classification. Every term is defined with the board exam in mind; you will find the key formula, the geometric meaning, and the exam context for each entry.

The terms are arranged alphabetically so you can use this list as a working reference during your review sessions. Do not study it passively. As you read each definition, ask yourself what type of board problem it generates, what formula it uses, and how it connects to the terms around it. Analytic geometry is a web of connected ideas, and recognizing those connections is what lets you solve unfamiliar problems by reasoning from first principles rather than hunting for a memorized formula.

Work through the conic sections with particular care. The circle, ellipse, parabola, and hyperbola together account for the largest share of analytic geometry problems on the board exam. Then give equal attention to the line formulas, the coordinate transformation techniques, and the three-dimensional extensions. These three areas separate reviewees who score average from those who score high. Use this list as your foundation and pair it with actual board exam problems for each topic cluster.

The 151 Analytic Geometry Terms and Definitions

1. Abscissa

The x-coordinate of a point in the rectangular coordinate system. In the ordered pair (x, y), the abscissa is always the first value. Board exams sometimes ask you to identify the abscissa of intersection points between curves, so do not confuse it with the ordinate, which is the y-coordinate.

2. Angle Between Two Lines

The acute angle θ formed by two intersecting lines, computed using tan θ = |(m₁ − m₂) / (1 + m₁m₂)|, where m₁ and m₂ are the slopes of the two lines. If the result is zero the lines are parallel. If the denominator is zero they are perpendicular. This formula appears frequently in board problems involving inclinations and intersections.

3. Angle Between Two Planes

The dihedral angle θ between two planes with normal vectors n₁ and n₂, found using cos θ = |n₁·n₂| / (|n₁||n₂|). The angle between the planes equals the angle between their normals (or its supplement, whichever is acute). Board problems on three-dimensional geometry often ask for this angle when two plane equations are given.

4. Angle of Inclination

The positive angle α that a line makes with the positive direction of the x-axis, measured counterclockwise, where 0° ≤ α < 180°. The slope and inclination are related by m = tan α. A line with inclination 45° has slope 1. A line with inclination 90° is vertical and has undefined slope.

5. Arc Length of a Polar Curve

The length of a curve defined in polar coordinates, given by L = ∫√(r² + (dr/dθ)²) dθ over the appropriate interval. For a cardioid r = a(1 + cos θ), the total arc length is 8a. Arc length of polar curves is a standard topic in calculus-based analytic geometry and appears in board problems that combine polar equations with integration.

6. Area Enclosed by a Polar Curve

The area bounded by a polar curve r = f(θ), given by A = (1/2)∫r² dθ over the interval [θ₁, θ₂]. For a full cardioid r = a(1 + cos θ), the enclosed area is (3/2)πa². This formula is tested in board problems that ask for the area inside a rose curve, limaçon, or lemniscate.

7. Area of a Triangle Using Coordinates

Given three vertices (x₁, y₁), (x₂, y₂), and (x₃, y₃), the area is A = ½|x₁(y₂ − y₃) + x₂(y₃ − y₁) + x₃(y₁ − y₂)|. This is the shoelace formula applied to a triangle. If the result is zero the three points are collinear. Board exams use this formula in locus problems, polygon area problems, and collinearity checks.

8. Asymptote of a Hyperbola

A pair of straight lines that the hyperbola approaches but never touches or crosses. For the standard hyperbola x²/a² − y²/b² = 1, the asymptotes are y = ±(b/a)x. They pass through the center of the hyperbola and are used for sketching. Board exams frequently ask for the equations of asymptotes given the hyperbola equation in general or standard form.

9. Axis of Symmetry

A line that divides a figure into two mirror-image halves. For a parabola y = ax² + bx + c, the axis of symmetry is the vertical line x = −b/(2a). For a horizontal parabola, the axis is horizontal. Ellipses and hyperbolas each have two axes of symmetry — the major and minor axes for the ellipse, and the transverse and conjugate axes for the hyperbola.

10. Cardioid

A heart-shaped polar curve defined by r = a(1 + cos θ) or r = a(1 + sin θ) depending on orientation. It is a special case of the limaçon where the two constants are equal. The cardioid passes through the pole and has a cusp there. Its enclosed area is (3/2)πa² and its total arc length is 8a. These values are standard board exam results.

11. Cartesian Coordinates

The rectangular coordinate system in which every point in the plane is identified by an ordered pair (x, y) representing signed distances from two perpendicular axes. The horizontal axis is the x-axis and the vertical axis is the y-axis. Their intersection is the origin (0, 0). In three dimensions, the system extends to ordered triples (x, y, z) with three mutually perpendicular axes.

12. Center of a Circle

The fixed point (h, k) equidistant from every point on the circle. The distance from the center to any point on the circle is the radius r. When the center is at the origin the equation simplifies to x² + y² = r². Identifying the center is the first step when working with circle problems on the board exam.

13. Center of a Hyperbola

The point (h, k) located exactly midway between the two vertices of a hyperbola. All key features namely foci, vertices, and asymptotes are defined relative to this center. For a standard hyperbola centered at the origin, h = k = 0.

14. Center of an Ellipse

The midpoint of both the major and minor axes of an ellipse, serving as the origin of the standard ellipse equation (x − h)²/a² + (y − k)²/b² = 1. The two foci and both pairs of vertices are symmetric about this center. Shifting the center away from the origin is handled by translation of axes.

15. Center of Curvature

The center of the osculating circle at a given point on a curve, located along the normal to the curve at a distance equal to the radius of curvature. The locus of all centers of curvature as a point moves along the curve is called the evolute. The center of curvature is relevant in kinematics and structural analysis of curved members.

16. Chord

A line segment whose two endpoints both lie on a curve, typically a circle or conic section. The longest chord of a circle is the diameter. Chords passing through the focus of a conic are called focal chords and carry special properties. The focal chord of a parabola perpendicular to the axis is called the latus rectum.

17. Circle

The set of all points in a plane equidistant from a fixed center point. The standard equation is (x − h)² + (y − k)² = r², where (h, k) is the center and r is the radius. The circle is a special case of an ellipse where both semi-axes are equal. Its eccentricity is exactly 0 and its discriminant in the general conic is B² − 4AC < 0 with A = C.

18. Circle — Angle in a Semicircle

The theorem that any angle inscribed in a semicircle is a right angle. Analytically, if (x₁, y₁) and (x₂, y₂) are the endpoints of a diameter and (x, y) is any other point on the circle, then (x − x₁)(x − x₂) + (y − y₁)(y − y₂) = 0. This equation of the circle using diameter endpoints is a direct encoding of the right-angle property and appears in board problems on loci.

19. Circle Through Three Points

Given any three non-collinear points, exactly one circle passes through all three. The center is found by solving two perpendicular bisector equations simultaneously. If the three points are collinear no such circle exists. This is a classic board exam problem type that tests the ability to set up and solve a 2×2 linear system from geometric conditions.

20. Cissoid

A plane curve defined in rectangular form as y²(2a − x) = x³, or in polar form as r = 2a sin θ tan θ. The cissoid has a cusp at the origin and an asymptote at x = 2a. It was historically used to solve the problem of doubling a cube. It appears occasionally in advanced curve-tracing and polar coordinate problems.

21. Coaxial Circles

A family of circles that share the same radical axis. Every pair of circles in a coaxial system has the same radical axis, and the centers of all circles in the system are collinear along a line perpendicular to the radical axis. Coaxial systems arise in problems involving families of circles and in the geometric analysis of circle pencils.

22. Collinear Points

Three or more points that all lie on the same straight line. To verify collinearity you can check that the slope between any two pairs of points is equal, or confirm that the area of the triangle they form equals zero using the shoelace formula. Collinearity tests appear in board problems involving locus equations and geometric constructions.

23. Common External Tangent

A line tangent to two circles that does not pass between them. For two circles with centers C₁ and C₂ and radii r₁ and r₂, the number of common external tangents depends on the relative positions of the circles. When the circles are external to each other, two common external tangents exist. The external center of similitude is the intersection point of the external tangents.

24. Common Internal Tangent

A line tangent to two circles that passes between them, touching each circle on the side facing the other. Common internal tangents exist only when the two circles do not overlap. Their intersection point is called the internal center of similitude. The distinction between internal and external tangents is tested in board problems involving two circles and their tangent configurations.

25. Completing the Square

An algebraic technique that rewrites a quadratic expression ax² + bx + c in the form a(x − h)² + k, identifying the vertex (h, k) of the parabola or the center and radius of a circle. It is the primary method for converting the general form of a conic to standard form. Proficiency in completing the square is essential for all conic section problems on the board exam.

26. Concurrent Lines

Three or more lines that all pass through a single common point. The point of concurrency is found by solving any two of the line equations simultaneously and verifying the result satisfies the third. Common examples include the medians, altitudes, and perpendicular bisectors of a triangle, all of which are concurrent at specific centers.

27. Condition for Tangency

The algebraic condition that a line is tangent to a conic that the system of equations formed by the line and conic has exactly one solution (discriminant = 0). For the line y = mx + c and the circle x² + y² = r², tangency requires c² = r²(1 + m²). This condition is derived by substituting the line into the conic equation and setting the discriminant of the resulting quadratic to zero.

28. Conic Section

A curve formed by the intersection of a plane with a right circular double cone. Depending on the angle of the cutting plane, the intersection is a circle, ellipse, parabola, or hyperbola. Degenerate cases include a point, a line, and a pair of intersecting lines. Conic sections are unified by the focus-directrix definition using eccentricity.

29. Conjugate Axis

The axis of a hyperbola perpendicular to the transverse axis, passing through the center. Its half-length is b, where b appears in the standard hyperbola equation. The conjugate axis does not pass through the vertices or foci but determines the shape of the asymptotes and the dimensions of the central rectangle used to sketch the hyperbola.

30. Conjugate Diameters

Two diameters of an ellipse or hyperbola that are conjugate to each other — each bisects chords parallel to the other. For an ellipse, conjugate diameters satisfy the condition m₁m₂ = −b²/a². The major and minor axes are the only conjugate diameters that are also perpendicular. Conjugate diameters appear in advanced conic problems and in the geometric construction of ellipses.

31. Conjugate Hyperbola

The hyperbola obtained by swapping the roles of a and b in a given hyperbola — that is, by using the conjugate axis as the new transverse axis. If the original hyperbola is x²/a² − y²/b² = 1, its conjugate is y²/b² − x²/a² = 1. The two hyperbolas share the same asymptotes, which is a useful property in curve sketching.

32. Coordinate Axes

The reference lines in a coordinate system. In the Cartesian plane, the x-axis is the horizontal reference and the y-axis is the vertical reference. In three-dimensional space, a third axis z is added perpendicular to both. The coordinate axes divide the plane into four quadrants and three-dimensional space into eight octants.

33. Coordinate Plane (3D)

Any of the three planes defined by pairs of coordinate axes in three-dimensional space: the xy-plane (z = 0), the xz-plane (y = 0), and the yz-plane (x = 0). Points in these planes satisfy one zero coordinate. The coordinate planes divide space into eight octants. Distance and locus problems in three-dimensional analytic geometry often involve these planes.

34. Curvature

A measure of how sharply a curve bends at a given point, defined as κ = |dy’/dx| / (1 + y’²)^(3/2) for a curve y = f(x), or equivalently κ = 1/R where R is the radius of curvature. High curvature means a tight bend. At an inflection point the curvature is zero. Curvature is used in kinematics, road design, and the mechanics of curved beams.

35. Degenerate Conic

A conic section that has collapsed into a simpler geometric object due to special conditions. The degenerate cases are a single point (an ellipse reduced to its center), a single line (a parabola with zero focal distance), and two intersecting lines (a hyperbola with a = 0 or b = 0). Degenerate conics arise when the discriminant conditions for a regular conic are only marginally satisfied.

36. Diameter of a Circle

A chord that passes through the center of the circle. Its length is 2r, where r is the radius. The diameter is the longest chord of a circle. In coordinate geometry, if the endpoints of a diameter are (x₁, y₁) and (x₂, y₂), the center is their midpoint and the equation of the circle can be written as (x − x₁)(x − x₂) + (y − y₁)(y − y₂) = 0.

37. Direction Cosines of a Line in Space

The cosines of the angles α, β, γ that a line makes with the positive x-, y-, and z-axes respectively: l = cos α, m = cos β, n = cos γ. They satisfy l² + m² + n² = 1. Direction cosines define the orientation of a line in three-dimensional space and are the components of the unit vector along the line. They appear in problems involving angles between lines and between a line and a plane.

38. Direction Numbers

Any set of three numbers (a, b, c) proportional to the direction cosines of a line in space. Direction numbers are not unique — any nonzero scalar multiple gives the same direction. They appear directly in the parametric and symmetric equations of a line. The direction vector of a line is (a, b, c), and the direction cosines are obtained by normalizing this vector.

39. Directrix

A fixed line used with the focus to define a conic section through the focus-directrix property. For any point on the conic, the ratio of its distance to the focus and its distance to the directrix equals the eccentricity e. The directrix of a parabola y² = 4px is x = −p. Every conic except the circle has one or two directrices.

40. Discriminant of the General Conic

For the general second-degree equation Ax² + Bxy + Cy² + Dx + Ey + F = 0, the discriminant is B² − 4AC. If B² − 4AC < 0 the conic is an ellipse (or circle if A = C and B = 0). If B² − 4AC = 0 it is a parabola. If B² − 4AC > 0 it is a hyperbola. This is the fastest classification tool for any general conic equation on the board exam.

41. Distance Between a Point and a Line

The perpendicular distance from a point (x₁, y₁) to the line Ax + By + C = 0, given by d = |Ax₁ + By₁ + C| / √(A² + B²). The absolute value ensures a non-negative result. This formula is tested in nearly every board exam and is used in problems involving tangent lines, minimum distances, and the definition of conics via focus-directrix.

42. Distance Between a Point and a Plane

The perpendicular distance from a point (x₁, y₁, z₁) to the plane Ax + By + Cz + D = 0, given by d = |Ax₁ + By₁ + Cz₁ + D| / √(A² + B² + C²). This is the three-dimensional analog of the point-to-line distance formula. It is used in problems involving parallel planes, perpendicular projections, and the geometry of solids in coordinate space.

43. Distance Between Parallel Lines

The perpendicular distance between two parallel lines Ax + By + C₁ = 0 and Ax + By + C₂ = 0, given by d = |C₁ − C₂| / √(A² + B²). The lines must have the same A and B coefficients (same normal direction) for this formula to apply. Board problems often require you to first rewrite the two equations in the same form before applying this formula.

44. Distance Between Two Points

The length of the segment connecting points (x₁, y₁) and (x₂, y₂), given by d = √((x₂ − x₁)² + (y₂ − y₁)²). In three dimensions, d = √((x₂ − x₁)² + (y₂ − y₁)² + (z₂ − z₁)²). This is the direct application of the Pythagorean theorem to coordinate geometry and is the most fundamental distance formula in the subject.

45. Distance Between Skew Lines

The shortest distance between two lines in three-dimensional space that do not intersect and are not parallel. It is computed as d = |( r₂ − r₁)·(d₁ × d₂)| / |d₁ × d₂|, where r₁ and r₂ are position vectors of points on each line and d₁ and d₂ are their direction vectors. The numerator is the scalar triple product of the connecting vector and the two direction vectors.

46. Distance from Origin to a Point

A special case of the distance formula where one point is the origin (0, 0): d = √(x² + y²). In three dimensions, d = √(x² + y² + z²). This equals the magnitude of the position vector of the point. It is used in converting between rectangular and polar coordinates and in problems involving circles centered at the origin.

47. Eccentricity

A non-negative scalar e that measures how much a conic deviates from being circular. For a circle e = 0, for an ellipse 0 < e < 1, for a parabola e = 1, and for a hyperbola e > 1. It is defined as e = c/a for ellipses and hyperbolas, where c is the focal distance and a is the semi-major or semi-transverse axis. Eccentricity is the single most important classification parameter for conics.

48. Eccentricity of a Conic in Polar Form

The value e appearing in the polar conic equation r = ed/(1 ± e cos θ) or r = ed/(1 ± e sin θ), which directly determines the type of conic. Extracting e from a given polar conic equation is a common board exam skill. The parameter d is the distance from the focus (at the pole) to the directrix, and ed equals the semi-latus rectum of the conic.

49. Equation of a Line in 3D — Parametric Form

A line through point (x₀, y₀, z₀) with direction vector (a, b, c) written as x = x₀ + at, y = y₀ + bt, z = z₀ + ct, where t is a scalar parameter. The parametric form is the most general representation of a line in three-dimensional space. It is used to find intersections with planes, distances between skew lines, and points of closest approach.

50. Equation of a Line in 3D — Symmetric Form

The symmetric form (x − x₀)/a = (y − y₀)/b = (z − z₀)/c obtained by eliminating the parameter t from the parametric equations. The constants a, b, c are the direction numbers of the line. If any direction number is zero, the corresponding variable is held constant and that equality is replaced by a separate equation for that variable.

51. Equation of a Plane

The general first-degree equation Ax + By + Cz + D = 0 representing a flat surface in three-dimensional space. The coefficients (A, B, C) form the normal vector to the plane. Given a point (x₀, y₀, z₀) on the plane and a normal vector (A, B, C), the equation is A(x − x₀) + B(y − y₀) + C(z − z₀) = 0. Plane equations are tested in board problems on distance, intersection, and angle between planes.

52. Equilateral Hyperbola

A hyperbola in which a = b, making the asymptotes perpendicular to each other. Also called a rectangular hyperbola. The standard form is x² − y² = a². In a rotated coordinate system, the rectangular hyperbola takes the form xy = k, which is the equation of an inverse proportion. The eccentricity of a rectangular hyperbola is always √2.

53. Evolute

The locus of all centers of curvature of a given curve. As a point moves along the original curve, the center of curvature traces the evolute. The original curve is called the involute of its evolute. For a parabola, the evolute is a semicubical parabola. The evolute is used in advanced curve analysis and in the design of gear tooth profiles.

54 Focal Chord

A chord of a conic that passes through a focus. For a parabola, the focal chord perpendicular to the axis is the latus rectum with length 4p. For an ellipse, the semi-latus rectum is b²/a. Focal chord properties are used in reflection problems — a ray from one focus of an ellipse reflects off the curve and passes through the other focus.

55. Focal Distance (c)

The distance from the center of a conic to each focus. For an ellipse, c² = a² − b². For a hyperbola, c² = a² + b². For a parabola, the single focus is at distance p from the vertex, where p is the focal parameter. The focal distance determines the eccentricity via e = c/a and controls how elongated or wide open the conic appears.

56. Focal Radii

The two line segments from the foci of an ellipse or hyperbola to a point on the curve. For an ellipse, the sum of the focal radii equals 2a (the major axis length). For a hyperbola, the difference of the focal radii equals 2a (the transverse axis length). These defining properties are the geometric definitions of the ellipse and hyperbola respectively.

57. Focus

A fixed point used with the directrix to define a conic through the focus-directrix ratio equal to eccentricity. A parabola has one focus. An ellipse and a hyperbola each have two foci. The focus is the reflective point — all rays parallel to the axis of a parabolic mirror converge at the focus, making this property fundamental in antenna, telescope, and solar concentrator design.

58. Four-Leaf Rose

A polar curve defined by r = a cos(2θ) or r = a sin(2θ), producing four equally spaced petals symmetric about both axes. Each petal has area πa²/8 and the total enclosed area is πa²/2. Rose curves with n petals are defined by r = a cos(nθ) — if n is even there are 2n petals, and if n is odd there are n petals.

59. General Equation of a Conic

The second-degree equation Ax² + Bxy + Cy² + Dx + Ey + F = 0. Every conic section (including degenerate cases) can be written in this form. The type of conic is identified by the discriminant B² − 4AC. Converting to standard form requires eliminating the xy term (by rotation when B ≠ 0) and completing the square to remove linear terms.

60. General Equation of a Circle

The form x² + y² + Dx + Ey + F = 0, obtained by expanding the standard form (x − h)² + (y − k)² = r². The center is (−D/2, −E/2) and the radius is r = √(D²/4 + E²/4 − F). A real circle requires D²/4 + E²/4 − F > 0. Board exams frequently give the general form and ask you to identify the center and radius by completing the square.

61. Generator of a Cone

A straight line on the surface of a cone that passes through the apex and lies entirely on the cone. The conic sections are named from the curves produced when a plane cuts the generator. Two generators at the apex meet at the vertex angle. The generator concept connects the algebraic definition of conics (via equations) to their original geometric definition as cone sections.

62. Hyperbola

The set of all points in a plane for which the absolute difference of the distances to two fixed points (foci) is constant and equal to 2a. The standard forms are x²/a² − y²/b² = 1 (horizontal) and y²/a² − x²/b² = 1 (vertical). The key relationship is c² = a² + b². The eccentricity satisfies e > 1 and the curve has two branches and two asymptotes.

63. Intercept Form of a Line

The equation x/a + y/b = 1, where a is the x-intercept and b is the y-intercept. This form is useful when two intercepts are given directly. It cannot represent lines through the origin or lines parallel to either axis. The intercept form is converted to general form by multiplying through by ab and rearranging.

64. Intersection of Two Lines

The unique point satisfying both line equations simultaneously, found by solving the two-equation linear system. If the system has no solution the lines are parallel. If it has infinitely many solutions the lines are identical. The intersection point is used in finding the vertices of polygons defined by line equations and in locating the pole with respect to a conic.

65. Intersection of Two Planes

The line formed when two non-parallel planes meet in three-dimensional space. Its direction vector is the cross product of the two normal vectors. A point on the line is found by solving the two plane equations simultaneously with one variable set to a convenient value. The intersection line is needed in problems involving the dihedral angle and the geometry of polyhedra.

66. Intercepts of a Plane

The points where a plane intersects the coordinate axes. For the plane x/a + y/b + z/c = 1, the intercepts are (a, 0, 0), (0, b, 0), and (0, 0, c). This intercept form of a plane equation is useful when the three intercepts are given. It fails for planes passing through the origin or parallel to a coordinate axis, which require the general form Ax + By + Cz + D = 0.

67. Involute

A curve traced by the end of a taut string as it unwinds from another curve (the evolute). For a circle of radius a, the involute has parametric equations x = a(cos θ + θ sin θ), y = a(sin θ − θ cos θ). Involutes of circles are used in the design of gear tooth profiles (involute gears) because they produce a constant pressure angle during meshing.

68. Latus Rectum

The focal chord of a conic perpendicular to the principal axis. For a parabola y² = 4px, the latus rectum has length 4p. For an ellipse, it has length 2b²/a. For a hyperbola, it has length 2b²/a. The latus rectum length is a direct measure of the width of the conic at the focus and is a standard value asked for in board problems on all conic sections.

69. Lemniscate of Bernoulli

A figure-eight polar curve defined by r² = a² cos(2θ) or r² = a² sin(2θ). The curve has two loops meeting at the pole and resembles an infinity symbol. Its total enclosed area is a². The lemniscate is the locus of points where the product of the distances from two fixed foci equals a²/4. It appears in curve identification and polar area problems.

70. Limaçon

A polar curve defined by r = a + b cos θ or r = a + b sin θ. When a > b the limaçon is a smooth convex or dimpled curve without a loop. When a = b it becomes a cardioid. When a < b it has an inner loop. The limaçon is one of the most commonly tested polar curves on the board exam because it includes the cardioid as a special case and produces a variety of distinct shapes.

151. y-Intercept

The point where a curve crosses the y-axis, where x = 0. To find it, substitute x = 0 into the equation. For the line y = mx + b, the y-intercept is b and can be read directly. For a conic in standard form, the y-intercept is found by setting x = 0 and solving the resulting equation. Knowing the y-intercept helps anchor a sketch to the coordinate axis.

CONCLUSION

Analytic geometry rewards reviewees who understand the underlying structure of the subject, not just individual formulas. The conic sections namely circle, parabola, ellipse, and hyperbola are unified by a single idea: the focus-directrix ratio equal to eccentricity. Every formula for foci, directrices, asymptotes, and latus rectum follows from that one definition. For the board exam, make sure you can move fluently between the general form and the standard form for all four conics, because board problems frequently give the general form and ask for properties that are only readable from the standard form. The discriminant B² − 4AC is your fastest classification tool and should be the first thing you compute when a general conic is presented.

For the line topics, the four formulas you must have completely memorized are the distance from a point to a line, the angle between two lines, the distance between parallel lines, and the midpoint and section formulas. These appear in almost every analytic geometry problem set on the board exam either directly or as intermediate steps. The rotation of axes and translation of axes techniques are less frequently tested but distinguish high scorers from average ones. You should know how to identify when a rotation is needed (nonzero B term) and how to find the angle. The three-dimensional extensions namely distance from a point to a plane, equations of lines in space, direction cosines, and the distance between skew lines are increasingly present on the board exam as the PRC curriculum has expanded its scope.

For polar coordinates and parametric equations, focus on the cardioid, limaçon, rose curves, and lemniscate as the most tested curve types. Know the area formula A = (1/2)∫r² dθ and the polar form of a general conic r = ed/(1 ± e cos θ). Parametric equations are most important for ellipses and circles, where they connect directly to trigonometric identities and are used in problems involving tangents and normals. Review this list systematically and work through each conic section cluster, then the line cluster, then the transformation cluster, and finally the three-dimensional cluster. Make sure to pair each group with actual board exam problems to lock in the application.

For practice problems on all these topics, head over to our Analytic Geometry Problems and Solutions section here on PinoyBix. Hundreds of solved exam-type questions, complete with step-by-step solutions, organized by topic so you can drill exactly what you need to work on.

Revisit the terms you found unfamiliar, work through sample board problems for each conic type, and you will be in solid shape for exam day. Good luck, and keep reviewing.