101 Analytic Geometry Terms Every Engineering Board Exam Taker Must Know

INTRODUCTION

If you are reviewing for the Philippine engineering licensure exam, analytic geometry is one of those topics you cannot afford to skip. It appears consistently across multiple engineering programs, including civil, electrical, electronics, mechanical, and others. The PRC board exam does not just test if you can memorize formulas. It tests if you actually understand what those formulas represent.

This list covers 101 terms you need to know. Each definition is written to be useful in two ways: first, as a quick review reference; second, as a conceptual foundation so you can actually solve problems under pressure. Whether you are a first-timer or a repeater brushing up on the fundamentals, this is a solid place to start.

Go through this list at your own pace. Mark the ones you are not confident about and come back to them. Analytic geometry rewards pattern recognition, and the more familiar these terms feel, the faster you will move through exam problems.

The 101 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.

2. Angle Between Two Lines

The acute angle θ formed by two intersecting lines. Computed using the formula: tan θ = |(m_₁ − m_₂) / (1 + m_₁m_₂)|, 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 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 135° has slope −1. This concept is equivalent to the inclination of a line, restated explicitly for clarity, as the PRC board exam uses both terms.

4. Area of a Triangle Using Coordinates

Given three vertices (x_₁, y_₁), (x_₂, y_₂), and (x_₃, y_₃), the area is: A = ½ |x_₁(y_₂ − y_₃) + x_₂(y_₃ − y_₁) + x_₃(y_₁ − y_₂)|. 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.

5. Asymptote of a Hyperbola

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

6. Axis of Symmetry

A line that divides a figure into two mirror-image halves. For a parabola in the form y = ax² + bx + c, the axis of symmetry is the vertical line x = −b/2a. For a horizontal parabola, the axis is horizontal. This term also applies to ellipses and hyperbolas, where both have two axes of symmetry.

7. Cardioid

A heart-shaped curve defined in polar coordinates by r = a(1 + cos θ) or r = a(1 + sin θ), depending on its orientation. It is a special case of the limaçon where a = b. The cardioid is formed by a point on a circle rolling around another circle of equal radius. Its total arc length and enclosed area are standard exam topics in calculus-based analytic geometry.

8. Center of a Circle

The fixed point (h, k) that is 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 usually the first step when working with circle problems on the board exam.

9. Center of a Hyperbola

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

10. Center of an Ellipse

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

11. Center of Curvature

The center of the osculating circle at a given point on a curve. It lies along the normal to the curve at that point, at a distance equal to the radius of curvature. The locus of all centers of curvature as a point moves along a curve forms a related curve called the evolute. In engineering, the center of curvature is relevant in kinematics, gear design, and structural analysis of curved members.

12. 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 that pass through the focus of a conic are called focal chords and carry special properties useful in board exam problems.

13. Circle

The set of all points in a plane that are 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 axes are equal. Its eccentricity is exactly 0.

14. Circle Through Three Points

Given any three non-collinear points, there exists exactly one circle that 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 (the “circle” degenerates to a line). This is a classic board exam problem type.

15. Cissoid

A curve defined in polar form as r = 2a sin θ tan θ, or equivalently, y² = x³ / (2a − x) in rectangular form. The cissoid was originally used in classical geometry to solve the problem of doubling a cube. It appears occasionally in advanced curve tracing problems and polar coordinate discussions.

16. Collinear Points

Three or more points that all lie on the same straight line. To verify collinearity, you can check if the slope between any two pairs of points is equal, or confirm that the area of the triangle they form equals zero. This concept shows up in board problems involving locus and line equations.

17. Conic Section

A curve formed by the intersection of a plane with a double-napped cone. The four standard conic sections are the circle, ellipse, parabola, and hyperbola. The type of conic depends on the angle of the intersecting plane relative to the cone. In analytic geometry, conics are identified by the general second-degree equation Ax² + Bxy + Cy² + Dx + Ey + F = 0.

18. Conjugate Axis

The axis of a hyperbola that is perpendicular to the transverse axis and passes through the center. Its length is 2b. The conjugate axis does not pass through the vertices of the hyperbola, but it is essential for drawing the reference rectangle used to construct the asymptotes.

19. Conjugate Hyperbola

A hyperbola formed by swapping the roles of a and b in the original hyperbola equation. If the original is x²/a² − y²/b² = 1, the conjugate hyperbola is y²/b² − x²/a² = 1. The two hyperbolas share the same asymptotes but open in perpendicular directions.

20. Curve Tracing

The systematic process of analyzing a curve’s key features: intercepts, symmetry, asymptotes, domain and range, and behavior at extremes, in order to sketch it accurately. Board exams sometimes ask you to identify properties of a curve from its equation without plotting individual points. Knowing the steps for curve tracing helps you answer those questions efficiently.

21. Degenerate Conic

A conic section that reduces to a simpler figure compose of a single point, a line, or a pair of intersecting lines rather than a proper curve. This happens when the discriminant and other coefficients take on special values. For example, x² + y² = 0 degenerates to a single point at the origin. Board exams may include degenerate cases as traps in multiple-choice problems.

22. Diameter of a Circle

A chord that passes through the center of the circle. It is the longest possible chord and equals twice the radius: d = 2r. The diameter also defines the angle-in-a-semicircle theorem: any inscribed angle subtending a diameter is always 90°. Board problems sometimes give the endpoints of a diameter and ask for the circle equation.

23. Directrix

A fixed line used along with the focus to define a parabola. Every point on the parabola is exactly equidistant from the focus and the directrix. For a vertical parabola with vertex at (h, k) and focal distance p, the directrix is the horizontal line y = k − p. For a horizontal parabola, it is the vertical line x = h − p.

24. Discriminant of the General Conic

The expression B² − 4AC from the general second-degree equation Ax² + Bxy + Cy² + Dx + Ey + F = 0. It is used to classify the conic without completing the square or rotating axes. Discriminant negative: ellipse or circle. Discriminant zero: parabola. Discriminant positive: hyperbola. This is the fastest classification tool for general conics and is a high-frequency board exam topic.

25. Distance Between Parallel Lines

The perpendicular distance between two parallel lines of the form Ax + By + C₁ = 0 and Ax + By + C₂ = 0. The formula is d = |C₁ − C₂| / √(A² + B²). Note that the coefficients A and B must be identical in both lines for this formula to apply. This is a direct formula you should have memorized before exam day.

26. Distance Formula

The straight-line distance between two points (x_₁, y_₁) and (x_₂, y_₂) in the plane: d = √[(x_₂ − x_₁)² + (y₂ − y₁)²]. This is derived from the Pythagorean theorem. It is one of the most fundamental formulas in analytic geometry and serves as the basis for circle and ellipse definitions.

27. Distance from a Point to a Line

The perpendicular distance from a point (x_₁, y_₁) to the line Ax + By + C = 0, given by d = |Ax₁ + By₁ + C| / √(A² + B²). The absolute value ensures the distance is always positive. Board problems use this formula in problems involving tangent lines, minimum distances, and locus problems.

28. Distance in Three Dimensions

The straight-line distance between two points (x_₁, y_₁, z_₁) and (x_₂, y_₂, z_₂) in 3D space: d = √[(x_₂ − x_₁)² + (y_₂ − y₁)² + (z_₂ − z₁)²]. This is a direct extension of the 2D distance formula. It is used in engineering applications involving spatial distances, structural lengths, and vector magnitudes.

29. Division of a Line Segment

The process of finding a point P that divides the line segment joining (x₁, y₁) and (x₂, y₂) in a given ratio m:n. For internal division: P = ((mx₂ + nx₁)/(m + n), (my₂ + ny₁)/(m + n)). For external division, the ratio is applied with a sign change. Division of a line segment problems appear in locus and centroid-related board exam questions.

30. Eccentricity

A numerical value e = c/a that describes the shape of a conic section. For a circle, e = 0. For an ellipse, 0 < e < 1. For a parabola, e = 1 exactly. For a hyperbola, e > 1. The closer e is to 0, the more circular the ellipse. As e increases beyond 1, the hyperbola opens wider. Eccentricity is frequently tested in board exams because it ties together all conic sections under one concept.

31. Ellipse

The set of all points in a plane where the sum of the distances from two fixed points (the foci) is constant and equal to 2a. The standard equation centered at the origin is x²/a² + y²/b² = 1, where a > b. The relationship between a, b, and c (focal distance) is c² = a² − b². Ellipses appear in orbital mechanics and structural engineering, but on board exams, they mostly appear as pure analytic geometry problems.

32. Equation of a Line

A mathematical relationship between x and y that is satisfied by every point on the line and no point off it. Lines can be written in several forms: slope-intercept (y = mx + b), point-slope (y − y_₁ = m(x − x_₁)), two-point form, intercept form, and general form (Ax + By + C = 0). Knowing how to convert between these forms quickly is a practical exam skill.

33. Equation of a Plane

A flat surface in three-dimensional space defined by the equation Ax + By + Cz + D = 0, where A, B, C are the components of a normal vector to the plane. Just as a line in 2D is defined by two points, a plane in 3D is defined by three non-collinear points or one point and a normal vector. Plane equations appear in engineering math courses as an introduction to multivariable geometry.

34. Focal Chord

A chord of a conic section that passes through one of its foci. In a parabola, the shortest focal chord is the latus rectum. In an ellipse or hyperbola, focal chords have specific length relationships tied to a, b, and c. Board problems involving focal chords usually ask for chord length or the coordinates of the endpoints.

35. Focus of a Hyperbola

One of two fixed points used to define a hyperbola. For the standard hyperbola x²/a² − y²/b² = 1, the foci are at (±c, 0) where c² = a² + b². Unlike the ellipse, the difference of distances from any point on the hyperbola to the two foci equals 2a. Notice that c is larger than a for hyperbolas, which means foci are always farther from the center than the vertices.

36. Focus of a Parabola

The fixed interior point used along with the directrix to define a parabola. For a vertical parabola with vertex at (h, k) and parameter p, the focus is at (h, k + p). For a horizontal parabola, the focus is at (h + p, k). The parabola opens toward the focus. All incoming rays parallel to the axis of symmetry reflect through the focus, which is why parabolic reflectors are used in satellite dishes and headlights.

37. Focus of an Ellipse

One of two fixed points inside an ellipse used to define the curve. For the standard ellipse x²/a² + y²/b² = 1, the foci are located at (±c, 0) where c² = a² − b². The sum of the distances from any point on the ellipse to the two foci always equals 2a. Board exams frequently ask for the coordinates of the foci given the ellipse equation.

38. General Equation of a Circle

The expanded form x² + y² + Dx + Ey + F = 0, obtained by expanding the standard circle equation. To find the center and radius, you complete the square on both x and y. If the result produces a negative value for r², the equation has no real solution. If r² = 0, it degenerates to a single point. Board exams regularly ask you to convert this form to standard form.

39. General Second-Degree Equation

The equation Ax² + Bxy + Cy² + Dx + Ey + F = 0, which can represent any conic section. The type of conic is determined by the discriminant B² − 4AC: if negative, it is an ellipse (or circle); if zero, it is a parabola; if positive, it is a hyperbola. The presence of the Bxy term indicates a rotation has occurred. This is one of the most concept-heavy topics in analytic geometry board problems.

40. Hyperbola

The set of all points in a plane where the absolute difference of distances to two fixed foci is constant and equal to 2a. The standard form is x²/a² − y²/b² = 1 (horizontal) or y²/a² − x²/b² = 1 (vertical). Unlike the ellipse, the hyperbola has two separate branches. The relationship among a, b, and c is c² = a² + b², which means c > a always.

41. Inclination of a Line

The angle α measured counterclockwise from the positive x-axis to the line, where 0° ≤ α < 180°. The slope m of the line equals tan α. A horizontal line has inclination 0° and slope 0. A vertical line has inclination 90° and undefined slope. Board problems sometimes give the inclination instead of the slope, so know the relationship well.

42. Intercept Form of a Line

The equation of a line written as x/a + y/b = 1, where a is the x-intercept and b is the y-intercept. This form is convenient when both intercepts are given directly. It cannot represent lines passing through the origin (since intercepts would both be zero) or vertical and horizontal lines with one intercept undefined.

43. Latus Rectum

The chord of a conic section that passes through a focus and is perpendicular to the principal axis (major axis for ellipses, axis of symmetry for parabolas). Its length differs by conic type. For a parabola, the latus rectum has length 4p. For an ellipse, it has length 2b²/a. For a hyperbola, it has length 2b²/a as well. Board exam problems on latus rectum appear regularly across all conic types.

44. Length of Latus Rectum (Ellipse)

For an ellipse with semi-major axis a and semi-minor axis b, the latus rectum length is 2b²/a. It represents the width of the ellipse measured through one of the foci, perpendicular to the major axis. This value is useful when you need to find specific chord lengths within an ellipse.

45. Length of Latus Rectum (Parabola)

Equal to 4p, where p is the distance from the vertex to the focus (also the distance from the vertex to the directrix). The latus rectum of a parabola tells you how wide the curve is at the level of the focus. A larger value of p means a wider, flatter parabola; a smaller p gives a narrower opening.

46. Limaçon

A polar curve of the form r = a + b cos θ or r = a + b sin θ. Its shape depends on the ratio a/b. When a = b, it becomes a cardioid. When a < b, it has an inner loop. When a > b, it is dimpled or convex. The name means “snail” in French, which fits its shape. Limaçons appear in polar coordinate problems and sometimes in area-calculation problems using integration.

47. Line

The set of all collinear points extending infinitely in both directions. In analytic geometry, a line is fully determined by two points or by one point and a slope. A line has a constant slope everywhere along it. Board exams test lines through various forms: slope-intercept, point-slope, two-point, intercept, and general form, and through application problems involving parallel and perpendicular conditions.

48. Linear Equation

An equation of degree 1, meaning neither variable appears with an exponent greater than 1 and there is no product of variables. Its graph is always a straight line. The general form is Ax + By + C = 0. Recognizing linearity quickly helps you identify the type of problem you are dealing with on the board exam.

49. Locus

The complete set of all points that satisfy a given geometric condition. Most definitions of curves in analytic geometry are locus definitions. For example, a circle is the locus of points equidistant from a center. A parabola is the locus of points equidistant from a focus and a directrix. Locus problems are common in the board exam because they test conceptual understanding, not just formula application.

50. Major Axis

The longest axis of an ellipse, passing through both foci and both vertices. Its total length is 2a, where a is the semi-major axis. For a horizontal ellipse, the major axis lies along the x-axis. For a vertical ellipse, it lies along the y-axis. Identifying which axis is major is the first step in setting up any ellipse problem.

101. y-Intercept

The point where a curve crosses the y-axis. At the y-intercept, the x-coordinate is zero. To find it, substitute x = 0 into the equation. For the line y = mx + b, the y-intercept is simply b and can be read directly from the equation. Knowing the y-intercept helps anchor a sketch of the curve to the coordinate axis.

CONCLUSION

That is all 101 terms. You will notice that many of these definitions are connected. The eccentricity ties together the circle, ellipse, parabola, and hyperbola; the focus appears in all conic definitions; slope is the backbone of every line topic. Studying these connections is more useful than memorizing each term in isolation.

For the Philippine engineering board exam, focus especially on the conic sections (Circle, Ellipse, Hyperbola, Parabola), the line formulas (Angle Between Two Lines, Distance Between Parallel Lines, Distance from a Point to a Line, Slope), the Discriminant of the General Conic, and the Translation of Axes technique. These topics recur the most frequently across actual board exam problems.

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.