Conic Sections Parabola Ellipse Hyperbola Questions for AGNIVEER NAVY

Parabola: Definition and Standard Forms

A parabola is the locus of all points equidistant from a fixed point (focus) and a fixed line (directrix). Think of it as a 'U-shaped' curve that opens in one direction. In Agniveer exams, you'll see parabolas in two standard orientations: opening right/left (horizontal axis) or up/down (vertical axis). The vertex is always at the origin in standard form, and the focus lies on the axis of symmetry at distance 'a' from the vertex.

• Parabola: distance to focus = distance to directrix
• Standard form y² = 4ax opens rightward; vertex at origin, focus at (a, 0)
• Standard form x² = 4ay opens upward; vertex at origin, focus at (0, a)
• Directrix for y² = 4ax is x = -a; for x² = 4ay is y = -a
• Eccentricity e = 1 for all parabolas (defining property)
• Latus rectum (focal chord perpendicular to axis) = 4a

Ellipse: Definition and Standard Forms

An ellipse is the locus of all points where the sum of distances to two fixed points (foci) is constant. It looks like a flattened circle. The major axis is the longer diameter, minor axis is shorter. Eccentricity e < 1 (always less than 1). In standard form, if major axis is horizontal, the equation is x²/a² + y²/b² = 1 where a > b. If major axis is vertical, swap a and b in the denominator positions.

• Ellipse: sum of distances to two foci = 2a (constant)
• Standard form (horizontal major axis): x²/a² + y²/b² = 1, where a > b
• Standard form (vertical major axis): x²/b² + y²/a² = 1, where a > b
• Foci distance from center: c = sqrt(a² - b²)
• Eccentricity e = c/a, always 0 < e < 1
• Latus rectum = 2b²/a

Hyperbola: Definition and Standard Forms

A hyperbola is the locus of all points where the difference of distances to two fixed points (foci) is constant. It has two separate branches opening away from each other. Eccentricity e > 1 (always greater than 1). Standard form depends on whether the transverse axis (the axis connecting the two branches) is horizontal or vertical. Asymptotes are straight lines the hyperbola approaches but never touches.

• Hyperbola: difference of distances to two foci = 2a (constant)
• Standard form (horizontal transverse): x²/a² - y²/b² = 1
• Standard form (vertical transverse): y²/a² - x²/b² = 1
• Foci distance from center: c = sqrt(a² + b²)
• Eccentricity e = c/a, always e > 1
• Asymptotes for x²/a² - y²/b² = 1 are y = ±(b/a)x

Identifying Conic Type from General Equation

When given a general second-degree equation Ax² + 2Hxy + By² + 2Gx + 2Fy + C = 0, you can identify the conic using the discriminant Delta = H² - AB. This is a quick classification trick that appears in 1-2 Agniveer questions. If Delta < 0, it's an ellipse (or circle). If Delta = 0, it's a parabola. If Delta > 0, it's a hyperbola. This method saves time in multiple-choice questions.

• Discriminant Delta = H² - AB
• Delta < 0: Ellipse (or circle if A = B and H = 0)
• Delta = 0: Parabola
• Delta > 0: Hyperbola
• This classification works for any orientation or position of the conic
• Useful for quick identification in MCQ format

• Confusing 4a with 2a: In parabola y² = 4ax, the focus is at distance a from vertex, NOT 2a. The coefficient is 4a, not 2a.
• Mixing up major and minor axes in ellipse: Always check which denominator is larger. If a² > b², then a is semi-major axis. Don't assume x-axis is always major.
• Forgetting c² = a² + b² for hyperbola vs c² = a² - b² for ellipse: Hyperbola uses PLUS (foci are farther out), ellipse uses MINUS (foci are inside).
• Asymptotes only for hyperbola: Parabolas and ellipses do NOT have asymptotes. Only hyperbolas do. This is a frequent distractor in MCQs.
• Eccentricity confusion: Parabola e = 1, Ellipse 0 < e < 1, Hyperbola e > 1. Many aspirants mix these up under time pressure.

• PEH Rule: Parabola e=1, Ellipse e<1, Hyperbola e>1. Remember as 'PEH' in order of increasing eccentricity.
• FOCI SPREAD: In ellipse, foci are INSIDE (c < a). In hyperbola, foci SPREAD OUT (c > a). Think of hyperbola as 'explosive' and ellipse as 'contained'.
• 4a in parabola: The '4' comes from the focal chord length = 4a. Memorize y² = 4ax as a unit.
• Asymptote = Hyperbola Only: 'A' for Asymptote, 'H' for Hyperbola. No asymptotes for P or E.

• Agniveer Navy SSR/MR papers typically ask 2-3 conic questions in 90-minute maths section. Expect 1 identification question, 1 focus/directrix question, and 1 property-based calculation.
• Standard form questions dominate: You'll rarely see rotated or shifted conics. Master the basic forms x²/a² ± y²/b² = 1 and y² = 4ax first.
• Eccentricity is a favorite: At least one question will ask for e or use e to identify the conic. Memorize the three ranges (0, 1, >1).
• Latus rectum appears occasionally: Know L = 4a for parabola and L = 2b²/a for ellipse. These are one-line calculations worth 1-2 marks.
• Time management: Conic questions are usually straightforward if you know the formulas. Spend max 3-4 minutes per question. If you're stuck on identification, use the discriminant method to save time.