CELE Engineering Mathematics — Analytic GeometryRevision Notes

Formulas

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

Section Title

Points, Distance, and Midpoint

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.

Formulas

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

Section Title

Straight Lines — Equations and Relationships

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.

Formulas

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

Section Title

Conic Sections — Circle

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.

Formulas

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

Section Title

Conic Sections — Parabola

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.

Formulas

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

Section Title

Conic Sections — Ellipse

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.

Formulas

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

Section Title

Conic Sections — 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.

Formulas

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

Section Title

Identifying Conics from the General Second-Degree Equation

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.