Analytic Geometry - Revision Notes
Analytic geometry bridges algebra and geometry by placing geometric figures on a coordinate system. For the PRC Civil Engineer Licensure Examination, this topic is a consistent source of board questions covering distances, lines, and conic sections. Mastery of this chapter is essential not only for the Mathematics subject but also for applications in surveying (traverse computations, road curve layout) and structural geometry. This revision note covers every examinable formula, worked board-style examples, common pitfalls, and exam strategies to maximize your score.
Points, Distance, and Midpoint
Formulas
d = sqrt((x2 - x1)^2 + (y2 - y1)^2)
Example
P1(1, 2), P2(4, 6): d = sqrt((4-1)^2 + (6-2)^2) = sqrt(9 + 16) = sqrt(25) = 5 units
Variables
d = distance between points P1(x1, y1) and P2(x2, y2)
Application
Finding the length of a line segment, the radius of a circle given center and a point, or verifying whether a triangle is isosceles or equilateral.
Midpoint M = ((x1 + x2)/2, (y1 + y2)/2)
Example
P1(1, 2), P2(4, 6): M = ((1+4)/2, (2+6)/2) = (2.5, 4)
Variables
M = midpoint coordinates; x1, y1, x2, y2 = coordinates of the two endpoints
Application
Finding the center of a segment, locating centroid components, verifying perpendicular bisectors.
Division point P = ((r1*x2 + r2*x1)/(r1 + r2), (r1*y2 + r2*y1)/(r1 + r2))
Example
Divide P1(0,0) to P2(6,9) in ratio 1:2: P = ((1*6 + 2*0)/3, (1*9 + 2*0)/3) = (2, 3)
Variables
r1:r2 = ratio in which P divides segment P1P2 internally
Application
Locating a point that divides a segment in a given ratio — common in board problems involving weighted centroids.
Exam tips
- Sketch a quick coordinate diagram to verify your numerical answer makes geometric sense.
- When a board problem asks 'find the length of the segment,' write the distance formula immediately without re-deriving it.
- For midpoint problems hidden as 'find the center of the segment,' the midpoint IS the center — apply the formula directly.
- Section formula: memorize it as a weighted average: P = (r1*P2 + r2*P1)/(r1 + r2).
Key points
- The Cartesian coordinate system locates every point P by an ordered pair (x, y) measured from the origin.
- Distance between two points is derived from the Pythagorean theorem applied to the horizontal and vertical legs of a right triangle.
- The midpoint divides a segment into two equal halves; use averaging of coordinates.
- Division of a line segment in ratio r1:r2 uses the section formula — a frequent board question variant.
- The distance formula is the foundation of all conic-section derivations; memorize it cold.
Definitions
Cartesian Coordinates
An ordered pair (x, y) representing the horizontal (abscissa) and vertical (ordinate) position of a point relative to a fixed origin and two perpendicular axes.
Importance
Foundation of all analytic geometry; every formula in this chapter is expressed in Cartesian terms.
Distance
The straight-line (Euclidean) length between two points, always a non-negative real number.
Importance
Used directly to find radii, segment lengths, and focal distances of conics.
Midpoint
The point equidistant from both endpoints of a line segment; computed by averaging the x- and y-coordinates separately.
Importance
Key in perpendicular bisector problems, center-finding, and segment division questions.
Common mistakes
- Subtracting coordinates in the wrong order inside the square — safe fix: always square the difference, sign does not matter.
- Forgetting to take the square root at the end of the distance formula.
- Confusing the section formula for internal vs. external division — external division uses a minus sign in the denominator.
- Applying the midpoint formula when the problem asks for a 1:3 division point instead of a 1:1 division.
Straight Lines — Equations and Relationships
Formulas
m = (y2 - y1) / (x2 - x1)
Example
Through (1,2) and (4,6): m = (6-2)/(4-1) = 4/3
Variables
m = slope; (x1, y1) and (x2, y2) = any two distinct points on the line
Application
Computing slope from two points; determining parallel/perpendicular relationships.
y - y1 = m(x - x1) [Point-slope form]
Example
m = 4/3, point (1, 2): y - 2 = (4/3)(x - 1) → 4x - 3y + 2 = 0
Variables
m = slope; (x1, y1) = known point on the line
Application
Writing the equation of a line when slope and one point are given — most efficient first step in most board problems.
y = mx + b [Slope-intercept form]
Example
4x - 3y + 2 = 0 → y = (4/3)x + 2/3; slope = 4/3, y-intercept = 2/3
Variables
m = slope; b = y-intercept
Application
Quickly reading slope and y-intercept from the equation; graphing.
Ax + By + C = 0 [General form]
Example
4x - 3y + 2 = 0: A = 4, B = -3, C = 2
Variables
A, B, C = real constants with A and B not both zero
Application
Standard form for computing point-to-line distance and finding intercepts.
d = |A*x0 + B*y0 + C| / sqrt(A^2 + B^2)
Example
Point (3,4), line 3x + 4y - 10 = 0: d = |3(3) + 4(4) - 10| / sqrt(9+16) = |9+16-10|/5 = 15/5 = 3 units
Variables
d = perpendicular distance; (x0, y0) = external point; Ax + By + C = 0 = line equation
Application
Finding the shortest distance from a point to a line — required for offset computations in surveying and road design.
d = |C1 - C2| / sqrt(A^2 + B^2) [Distance between parallel lines Ax+By+C1=0 and Ax+By+C2=0]
Example
3x+4y-10=0 and 3x+4y+5=0: d = |-10-5|/sqrt(9+16) = 15/5 = 3 units
Variables
C1, C2 = constant terms of the two parallel lines (A and B coefficients must be identical)
Application
Finding the distance between two parallel lines — a common board question variant.
tan(α) = |(m1 - m2) / (1 + m1*m2)|
Example
m1 = 2, m2 = 1/3: tan(α) = |(2 - 1/3)/(1 + 2*1/3)| = |(5/3)/(5/3)| = 1 → α = 45°
Variables
α = acute angle between two lines; m1, m2 = their respective slopes
Application
Finding the angle of intersection between two lines.
Exam tips
- Always rearrange the line equation to Ax + By + C = 0 before applying the distance formula — this is a one-step setup that prevents errors.
- For perpendicular bisector questions: (1) find the midpoint, (2) find the slope of the given segment, (3) use negative reciprocal as the new slope, (4) apply point-slope form.
- When the board problem says 'find the distance between the lines,' first verify they are parallel (same A and B ratios), then apply the parallel-line distance formula directly.
- The acute angle formula fails when 1 + m1*m2 = 0, which means the lines are perpendicular (α = 90°) — treat this as a special case.
Key points
- Slope m measures steepness and direction: positive slope rises left-to-right, negative slope falls.
- Parallel lines have equal slopes; perpendicular lines have slopes that are negative reciprocals of each other.
- Three standard forms of a line equation: point-slope, slope-intercept, and general (Ax + By + C = 0).
- The angle of inclination θ satisfies m = tan(θ), where 0° ≤ θ < 180°.
- The acute angle between two intersecting lines uses the formula tan(α) = |(m1 - m2)/(1 + m1*m2)|.
- Distance from a point to a line is one of the highest-frequency analytic geometry questions on the board exam.
Definitions
Slope (m)
The tangent of the angle of inclination; numerically equal to the rise over the run between any two points on the line.
Importance
The single most important property of a straight line; determines parallelism and perpendicularity.
Perpendicular Lines
Two lines are perpendicular if and only if the product of their slopes equals -1: m1 × m2 = -1.
Importance
Frequently tested in problems asking for a line normal to a given line or the perpendicular bisector of a segment.
x-intercept and y-intercept
The x-intercept is where the line crosses the x-axis (set y = 0); the y-intercept is where it crosses the y-axis (set x = 0).
Importance
Used in intercept form x/a + y/b = 1 and in finding bounding coordinates.
General Form
Ax + By + C = 0, the form required to apply the point-to-line distance formula directly.
Importance
Always convert the line equation to general form before using the distance formula to avoid sign errors.
Common mistakes
- Perpendicular slope — students often negate the slope without taking the reciprocal. The perpendicular slope of m is -1/m, not -m.
- Sign error in the point-to-line distance — omitting the absolute value gives a negative distance, which is physically impossible.
- For parallel line distance, students sometimes use different A and B coefficients. The coefficients must be identical (same form) before subtracting C values.
- Confusing the angle between two lines (acute angle formula) with the inclination angle (measured from positive x-axis).
- Using the slope formula when the line is vertical (undefined slope) — vertical lines have the equation x = constant.
Conic Sections — Circle
Formulas
(x - h)^2 + (y - k)^2 = r^2 [Standard form]
Example
Center (2, -3), point (5, 1): r = sqrt((5-2)^2 + (1-(-3))^2) = sqrt(9+16) = 5; equation: (x-2)^2 + (y+3)^2 = 25
Variables
(h, k) = center; r = radius
Application
Writing the circle equation when center and radius (or center and a point on the circle) are given.
x^2 + y^2 + Dx + Ey + F = 0 [General form]
Example
x^2 + y^2 - 4x + 6y - 12 = 0: center = (2, -3), r = sqrt(4 + 9 + 12) = 5
Variables
D, E, F = constants; center = (-D/2, -E/2); r = sqrt((D/2)^2 + (E/2)^2 - F)
Application
Reading center and radius from a circle equation not in standard form — complete the square or use center formula directly.
Exam tips
- When given a circle equation in general form, use the shortcut center = (-D/2, -E/2) and r = sqrt((D/2)^2 + (E/2)^2 - F) to save time.
- To verify that three points are on the same circle, substitute all three into the general form and solve the 3×3 system for D, E, F.
- Board problems sometimes give the endpoints of a diameter — find the center (midpoint) and radius (half the diameter) directly.
Key points
- A circle is the locus of all points equidistant from a fixed center point; the constant distance is the radius r.
- Standard form directly shows the center and radius for quick identification.
- General second-degree form Ax^2 + Ay^2 + Dx + Ey + F = 0 (equal coefficients for x^2 and y^2, no xy term) represents a circle.
- Completing the square converts general form to standard form — a critical algebraic skill for board problems.
- Eccentricity of a circle is e = 0.
Definitions
Circle
Locus of points equidistant (distance = r) from a fixed point called the center.
Importance
Most straightforward conic; frequently appears as a warm-up or setup problem on the board exam.
Completing the Square
Algebraic technique of adding (b/2)^2 to both sides of a quadratic expression to form a perfect square trinomial.
Importance
The primary method for converting general conic form to standard form for all four conics.
Common mistakes
- Forgetting to square the radius — the right-hand side of the standard form is r^2, not r.
- Sign error when reading the center: (x - h)^2 means center x-coordinate is +h, not -h.
- In completing the square, adding the completing term to the left side but forgetting to add it to the right side as well.
- Misidentifying a circle when the x^2 and y^2 coefficients are not equal — those represent ellipses.
Conic Sections — Parabola
Formulas
(y - k)^2 = 4a(x - h) [Horizontal parabola, opens right if a > 0]
Example
Vertex (1, 2), focus (3, 2): a = 3-1 = 2; equation: (y-2)^2 = 8(x-1)
Variables
(h, k) = vertex; a = focal distance; focus at (h+a, k); directrix x = h - a
Application
Parabolas with horizontal axis of symmetry — common in arch and cable problems.
(x - h)^2 = 4a(y - k) [Vertical parabola, opens upward if a > 0]
Example
Vertex (0, 0), focus (0, 3): a = 3; equation: x^2 = 12y
Variables
(h, k) = vertex; a = focal distance; focus at (h, k+a); directrix y = k - a
Application
Parabolas with vertical axis — satellite dish, projectile path approximations.
Exam tips
- The quick rule: the variable that is NOT squared tells you the axis direction and which way the parabola opens.
- Always extract a from the equation first: set 4a equal to the coefficient, then find a = coefficient/4.
- Focus is always INSIDE the parabola; directrix is always OUTSIDE — use this to check your answer.
- For board problems asking 'what is the equation of the directrix,' it is always the line at distance a from the vertex, opposite the focus.
Key points
- A parabola is the locus of points equidistant from a fixed point (focus) and a fixed line (directrix).
- The vertex is the midpoint between the focus and the directrix.
- The focal distance a is the distance from the vertex to the focus (and also from the vertex to the directrix).
- Four standard orientations: opens right, left, upward, downward.
- The latus rectum (chord through the focus perpendicular to the axis) has length 4a.
- Eccentricity of a parabola is exactly e = 1.
Definitions
Focus
The fixed point inside the parabola from which all points on the parabola are equidistant to the directrix.
Importance
Determines the shape and orientation; a = distance from vertex to focus.
Directrix
The fixed line outside the parabola; every point on the parabola is equidistant from the focus and this line.
Importance
Used to derive the parabola equation and to verify the eccentricity definition.
Latus Rectum
The chord of the parabola that passes through the focus and is perpendicular to the axis of symmetry; its length = 4a.
Importance
Board problems sometimes ask for the length of the latus rectum — read the coefficient of the variable directly: 4a = coefficient, so a = coefficient/4.
Common mistakes
- Misidentifying a as the coefficient instead of a = coefficient/4. If (y-k)^2 = 12(x-h), then 4a = 12, so a = 3, not a = 12.
- Confusing which variable is squared to determine the axis of symmetry — squared y means horizontal axis, squared x means vertical axis.
- Placing the focus in the wrong direction — if a is negative, the parabola opens in the opposite direction.
- Forgetting that the directrix is on the OPPOSITE side of the vertex from the focus, at distance a.
Conic Sections — Ellipse
Formulas
(x-h)^2/a^2 + (y-k)^2/b^2 = 1 [Horizontal major axis, a > b]
Example
x^2/25 + y^2/9 = 1: a = 5, b = 3, c = sqrt(25-9) = 4; foci at (±4, 0); e = 4/5 = 0.8
Variables
(h, k) = center; a = semi-major axis along x; b = semi-minor axis along y; c^2 = a^2 - b^2; foci at (h ± c, k)
Application
Most common ellipse orientation in board problems — major axis along x-axis.
(x-h)^2/b^2 + (y-k)^2/a^2 = 1 [Vertical major axis, a > b]
Example
x^2/9 + y^2/25 = 1: a = 5 (under y^2), b = 3, c = 4; foci at (0, ±4)
Variables
(h, k) = center; a = semi-major axis along y; b = semi-minor axis along x; foci at (h, k ± c)
Application
Ellipse with major axis along the y-axis — identify by the larger denominator being under y^2.
c^2 = a^2 - b^2 [Ellipse focal relation]
Example
a = 5, b = 3: c = sqrt(25 - 9) = sqrt(16) = 4
Variables
a = semi-major axis; b = semi-minor axis; c = center-to-focus distance
Application
Finding the focal distance whenever a and b are known.
e = c/a [Eccentricity]
Example
c = 4, a = 5: e = 4/5 = 0.8 (ellipse, moderately elongated)
Variables
e = eccentricity (0 < e < 1 for ellipse); c = focal distance; a = semi-major axis
Application
Classifying the conic and measuring how 'elongated' the ellipse is.
Exam tips
- Quick check: in the standard form, the LARGER denominator gives a^2. Identify this first, then compute c^2 = a^2 - b^2.
- The foci are always on the major axis (the axis with the larger denominator) — this determines whether to write foci as (h±c, k) or (h, k±c).
- For 'find the eccentricity' questions: compute c first, then divide by a. The answer must be between 0 and 1 for an ellipse.
- Area of an ellipse = πab — a bonus formula sometimes tested in the board exam.
Key points
- An ellipse is the locus of points for which the SUM of distances to two fixed points (foci) is constant and equal to 2a.
- Semi-major axis a > semi-minor axis b; the foci lie along the major axis.
- Key relationship: c^2 = a^2 - b^2, where c is the distance from center to each focus.
- Eccentricity e = c/a; for an ellipse, 0 < e < 1.
- A circle is a special ellipse with a = b, c = 0, and e = 0.
- The sum of distances from any point on the ellipse to both foci always equals 2a.
Definitions
Semi-major Axis (a)
Half the longest diameter of the ellipse; always the larger of the two semi-axes. Always assign a to the larger denominator's square root.
Importance
Determines the size and the constant sum of focal distances (2a). Critical to identify correctly before computing c.
Semi-minor Axis (b)
Half the shortest diameter of the ellipse; b < a always.
Importance
Used in the focal relation c^2 = a^2 - b^2 and in computing the area (πab).
Eccentricity (e)
Ratio e = c/a measuring the deviation from circularity. For an ellipse: 0 < e < 1.
Importance
Board problems frequently ask to compute e or to classify a conic using e.
Common mistakes
- Using c^2 = a^2 + b^2 (hyperbola formula) instead of c^2 = a^2 - b^2 for an ellipse — this is the single most common conic error on the board exam.
- Assigning a to the x-denominator regardless of which is larger — always assign a to the LARGER denominator.
- Computing e = b/a or e = a/c instead of e = c/a.
- Forgetting that foci lie along the MAJOR axis — if the larger denominator is under y^2, foci move along the y-axis.
Conic Sections — Hyperbola
Formulas
(x-h)^2/a^2 - (y-k)^2/b^2 = 1 [Horizontal transverse axis]
Example
x^2/16 - y^2/9 = 1: a = 4, b = 3, c = sqrt(16+9) = 5; foci (±5, 0); e = 5/4 = 1.25; asymptotes y = ±(3/4)x
Variables
(h, k) = center; a = semi-transverse axis; b = semi-conjugate axis; c^2 = a^2 + b^2; foci at (h±c, k); asymptotes: y - k = ±(b/a)(x - h)
Application
Hyperbola opening left and right — identified by positive x^2 term.
(y-k)^2/a^2 - (x-h)^2/b^2 = 1 [Vertical transverse axis]
Example
y^2/25 - x^2/16 = 1: a = 5, b = 4, c = sqrt(25+16) = sqrt(41); foci at (0, ±sqrt(41))
Variables
(h, k) = center; a = semi-transverse axis along y; foci at (h, k±c); asymptotes: y - k = ±(a/b)(x - h)
Application
Hyperbola opening up and down — identified by positive y^2 term.
c^2 = a^2 + b^2 [Hyperbola focal relation]
Example
a = 4, b = 3: c = sqrt(16 + 9) = sqrt(25) = 5
Variables
a = semi-transverse axis; b = semi-conjugate axis; c = center-to-focus distance
Application
Finding c whenever a and b are known for a hyperbola.
Asymptotes: y - k = ±(b/a)(x - h) [For horizontal hyperbola]
Example
x^2/16 - y^2/9 = 1: asymptotes y = ±(3/4)x
Variables
Slope = ±b/a; (h, k) = center of the hyperbola
Application
Graphing the hyperbola and answering questions about limiting behavior.
Exam tips
- Memory trick for focal relations: Ellipse = c^2 = a^2 MINUS b^2; Hyperbola = c^2 = a^2 PLUS b^2. The sign in the equation matches the focal relation: minus in equation → subtract for c; mixed signs → add for c.
- The positive term in the standard form tells you which direction the hyperbola opens and where the vertices are.
- For asymptote equations, the slopes are ±b/a (horizontal) or ±a/b (vertical) — always check which form you have.
- Eccentricity e > 1 is the guaranteed identifier of a hyperbola — compute e = c/a and verify it exceeds 1.
Key points
- A hyperbola is the locus of points for which the DIFFERENCE of distances to two foci is constant and equal to 2a.
- Key relationship: c^2 = a^2 + b^2 (note the PLUS sign, opposite of the ellipse).
- Eccentricity e = c/a > 1 for all hyperbolas.
- Hyperbolas have two asymptotes passing through the center with slopes ±b/a (horizontal form) or ±a/b (vertical form).
- The transverse axis connects the two vertices; the conjugate axis is perpendicular to it.
- A rectangular hyperbola has a = b, giving asymptotes at 45° angles.
Definitions
Transverse Axis
The axis of the hyperbola along which the two vertices and two foci lie; has length 2a.
Importance
Determines the orientation (horizontal or vertical) of the hyperbola.
Asymptote
Lines that the hyperbola approaches but never touches as it extends to infinity.
Importance
Board problems may ask for the equation of the asymptote — it passes through the center with slope ±b/a.
Eccentricity (e > 1)
Ratio e = c/a; for a hyperbola, e is always greater than 1 because c > a always holds (from c^2 = a^2 + b^2).
Importance
The defining criterion for a hyperbola — if e > 1, it is a hyperbola.
Common mistakes
- Using c^2 = a^2 - b^2 (ellipse formula) for a hyperbola — always remember: HYPERBOLA ADDS, ELLIPSE SUBTRACTS.
- Assigning a to the first denominator regardless of which term is positive — for hyperbola, a always goes with the POSITIVE term.
- Writing asymptote slopes as ±a/b instead of ±b/a for horizontal hyperbolas (or vice versa for vertical).
- Confusing the conjugate axis b with the focal distance c.
Identifying Conics from the General Second-Degree Equation
Formulas
B^2 - 4AC: < 0 → ellipse/circle; = 0 → parabola; > 0 → hyperbola
Example
9x^2 + 4y^2 - 36 = 0: A=9, B=0, C=4; B^2-4AC = 0 - 4(9)(4) = -144 < 0 → ellipse
Variables
A, B, C = coefficients in Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0
Application
Rapidly classifying any second-degree equation without completing the square.
Exam tips
- For PRC board problems, B = 0 almost always (axes parallel to coordinate axes), so the A vs. C comparison is sufficient and fast.
- When in doubt, complete the square and bring it to standard form — this is slower but guaranteed to give the correct identification.
Key points
- The general second-degree equation: Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0.
- For conics with axes parallel to the coordinate axes (no Bxy term, B = 0), identification uses the coefficients of x^2 and y^2.
- Circle: A = C (equal and same sign coefficients for x^2 and y^2, no xy term).
- Ellipse: A ≠ C but both same sign (both positive after factoring).
- Hyperbola: A and C have opposite signs.
- Parabola: exactly one of A or C equals zero.
- The discriminant B^2 - 4AC classifies the conic when B ≠ 0: < 0 ellipse/circle, = 0 parabola, > 0 hyperbola.
Definitions
Discriminant of a Conic
The expression B^2 - 4AC used to classify the type of conic section without completing the square.
Importance
A powerful shortcut for identification questions on the board exam.
Common mistakes
- Checking only the presence of x^2 and y^2 without ensuring their coefficients are equal for a circle — 2x^2 + 2y^2 = 8 is still a circle, but 2x^2 + 3y^2 = 8 is an ellipse.
- Forgetting to account for the cross term Bxy — if B ≠ 0, the conic axes are tilted and the simple A vs. C comparison is not sufficient.
Connections
- Distance formula → Circle equation: The circle is defined as the locus of points at distance r from the center; substituting the distance formula directly yields the standard form.
- Completing the square → All conics: This single algebraic technique converts general form to standard form for circles, ellipses, parabolas, and hyperbolas — master it once, apply everywhere.
- Point-to-line distance → Parabola definition: The parabola's definition (equidistant from focus and directrix) is essentially two applications of the distance formula set equal to each other.
- Eccentricity unifies all conics: e = 0 (circle), 0 < e < 1 (ellipse), e = 1 (parabola), e > 1 (hyperbola) — one parameter classifies four curves.
- Focal relations: c^2 = a^2 - b^2 (ellipse, subtract) vs. c^2 = a^2 + b^2 (hyperbola, add) — the sign of the relationship mirrors the sign between the two squared terms in the standard form equation.
- Slope relationships: m1 × m2 = -1 (perpendicular) connects to the normal line in calculus and to the perpendicular bisector in geometry — the same formula applies throughout engineering mathematics.
- Surveying application: Traverse coordinate computations use the distance and midpoint formulas extensively; the point-to-line distance formula gives perpendicular offsets in road alignment design.
- Structural geometry: Parabolic cable shapes in suspension bridges, elliptical arches in bridge design, and hyperbolic cooling towers — all governed by the conic equations in this chapter.
Quick review questions
Find the distance between the points A(3, -1) and B(-2, 11).
Answer
d = 13 units
d = sqrt((-2-3)^2 + (11-(-1))^2) = sqrt((-5)^2 + (12)^2) = sqrt(25 + 144) = sqrt(169) = 13 units. Note the 5-12-13 Pythagorean triple.
Find the equation of the line perpendicular to 3x - 4y + 8 = 0 and passing through the point (6, 1).
Answer
4x + 3y - 27 = 0
Slope of given line: m1 = 3/4 (from 3x - 4y + 8 = 0 → y = (3/4)x + 2). Perpendicular slope: m2 = -4/3. Point-slope form: y - 1 = (-4/3)(x - 6) → 3(y-1) = -4(x-6) → 3y - 3 = -4x + 24 → 4x + 3y - 27 = 0.
Find the center and radius of the circle defined by x^2 + y^2 - 6x + 4y - 12 = 0.
Answer
Center (3, -2), radius r = 5
Use the shortcut: center = (-D/2, -E/2) = (6/2, -4/2) = (3, -2). Radius r = sqrt((D/2)^2 + (E/2)^2 - F) = sqrt(9 + 4 - (-12)) = sqrt(9 + 4 + 12) = sqrt(25) = 5.
Given the ellipse x^2/25 + y^2/9 = 1, find (a) the foci, (b) the eccentricity, and (c) the length of the latus rectum.
Answer
(a) Foci at (±4, 0); (b) e = 4/5 = 0.8; (c) LR = 2b^2/a = 2(9)/5 = 3.6 units
a^2 = 25 → a = 5; b^2 = 9 → b = 3. c^2 = a^2 - b^2 = 25 - 9 = 16 → c = 4. Foci at (±4, 0) since major axis is along x. Eccentricity e = c/a = 4/5 = 0.8. Length of latus rectum = 2b^2/a = 2(9)/5 = 18/5 = 3.6 units.
Identify the conic and find its key features: 9x^2 - 16y^2 = 144.
Answer
Hyperbola; center (0,0); a = 4, b = 3, c = 5; e = 5/4 = 1.25; foci at (±5, 0); asymptotes y = ±(3/4)x
Divide by 144: x^2/16 - y^2/9 = 1. Positive x^2 term → horizontal hyperbola. a^2 = 16 → a = 4; b^2 = 9 → b = 3. c^2 = a^2 + b^2 = 16 + 9 = 25 → c = 5. e = 5/4 = 1.25 > 1 confirms hyperbola. Foci at (±5, 0). Asymptotes: y = ±(b/a)x = ±(3/4)x.
The parabola (y - 2)^2 = -8(x + 1) has its vertex at (-1, 2). Find the focus and directrix.
Answer
Focus at (-3, 2); directrix x = 1
Comparing with (y-k)^2 = 4a(x-h): 4a = -8, so a = -2 (negative means opens left). Vertex (h, k) = (-1, 2). Focus at (h + a, k) = (-1 + (-2), 2) = (-3, 2). Directrix: x = h - a = -1 - (-2) = 1.
Find the distance between the parallel lines 4x - 3y + 10 = 0 and 4x - 3y - 15 = 0.
Answer
d = 5 units
Both lines have the same A = 4 and B = -3. C1 = 10, C2 = -15. d = |C1 - C2| / sqrt(A^2 + B^2) = |10 - (-15)| / sqrt(16 + 9) = 25 / sqrt(25) = 25/5 = 5 units.
Diagrams
Analytic Geometry — Complete Chapter Overview
A comprehensive mind map showing the hierarchical structure of the entire Analytic Geometry chapter, branching from Points and Lines into formulas and relationships, and from Conic Sections into the four conics with their key properties and eccentricities.
Conic Section Identification Flowchart
A decision-tree flowchart guiding the student through systematic identification of any conic section from its general second-degree equation. The tree first checks for a cross term (xy), then compares coefficients of x-squared and y-squared to determine the conic type, ending with the appropriate solution procedure for each type.
Step-by-Step Problem-Solving Guide for Conic Sections
A step-by-step flowchart showing the universal problem-solving procedure for any conic section board problem. The chart guides students from reading the problem, through identifying the conic, extracting parameters, computing the focal relation, and verifying the answer before writing the final solution.
Focal Relations — Ellipse vs Hyperbola Memory Guide
A focused flowchart contrasting the ellipse and hyperbola focal relations side-by-side. The diagram highlights the critical difference — ellipse subtracts (c-squared = a-squared minus b-squared) while hyperbola adds (c-squared = a-squared plus b-squared) — with memory tricks to prevent the most common board-exam error in this topic.
Distribution of Analytic Geometry Board Exam Questions by Topic
A pie chart showing the approximate distribution of PRC board exam questions across the five major topics in Analytic Geometry. Points and Lines accounts for the largest share (30%), followed by equal distribution among the three major conics (20% each), with Hyperbolas at 10%. Use this to prioritize your review time.
Exam strategy
On the PRC Civil Engineer board exam, Analytic Geometry questions typically number 5–10 items per sitting and are distributed across points/lines (30%), circles (20%), parabolas (20%), ellipses (20%), and hyperbolas (10%). Follow this attack strategy: (1) READ the question completely and identify which conic or line concept is being tested. (2) WRITE the standard form equation immediately — do not proceed without it. (3) EXTRACT the parameters (h, k, a, b, c) by inspection or by completing the square. (4) APPLY the requested formula directly. (5) CHECK that your answer is dimensionally and geometrically consistent (e.g., eccentricity of an ellipse must be between 0 and 1; radius must be positive). For multiple-choice items, plug your answer back into the original equation to verify. Manage time by mastering the shortcut formulas (center from general circle form, focal relations) so you spend no more than 90 seconds per item. The highest-value study investments are: (a) the point-to-line distance formula, (b) the ellipse focal relation c^2 = a^2 - b^2 versus the hyperbola c^2 = a^2 + b^2, and (c) extracting a from the parabola coefficient 4a. These three concepts together account for roughly 50% of all Analytic Geometry board questions.