What is Hyperbola in CUET UG?

Hyperbola is a topic in the Mathematics syllabus of CUET UG. It carries a weight of 3% in the exam. Our notes cover the key concepts, formulas, and problem-solving approaches you need to master this topic.

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Hyperbola

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision.

Hyperbola — Key Facts for CUET • Standard Form: x²/a² - y²/b² = 1 (transverse axis horizontal); y²/a² - x²/b² = 1 (transverse axis vertical) • Most tested CUET concept: Finding equation of hyperbola given focus and directrix or eccentricity • Common mistake students make: Confusing ‘a² + b² = c²’ (ellipse property) with ‘c² = a² + b²’ (hyperbola property) — students often mix these up • Key technique or method to solve quickly: Use relationship c² = a² + b² and e = c/a; given two parameters, find the third immediately • Important exception or special case: Rectangular hyperbola has asymptotes perpendicular (a = b), equation xy = c² • Most frequent question type in CUET: Given eccentricity and focus, find equation; or find coordinates of foci/vertices given standard equation ⚡ Exam tip: When asked for length of latus rectum, remember it’s always 4a²/b for hyperbola (not 2a like ellipse). Also, if equation has minus sign between x² and y² terms, that’s your hyperbola — ellipse has only plus signs.

🟡 Standard — Regular Study (2d–2mo)

Standard content for students with a few days to months.

Hyperbola — CUET Study Guide

A hyperbola is the set of all points in a plane where the absolute difference of distances from two fixed points (foci) is constant and equals 2a. The standard equation with center at origin is x²/a² - y²/b² = 1, where the transverse axis lies along the x-axis. Here, c² = a² + b², where c is the distance from center to each focus, and eccentricity e = c/a > 1.

The key properties include vertices at (±a, 0), co-vertices at (0, ±b), and asymptotes given by y = ±(b/a)x. The latus rectum has endpoints at (c, ±b²/a) and length 4b²/a. For CUET, remember that e > 1 distinguishes hyperbolas from ellipses.

Typical CUET patterns: Questions often give eccentricity and a focus, requiring you to find the equation. Always verify which axis the transverse axis lies on — if the negative term involves x², transverse axis is horizontal; if y², it’s vertical.

Common traps: Watch for “conjugate hyperbola” questions — x²/a² - y²/b² = -1 swaps the signs. Also, parametric form (a secθ, b tanθ) is frequently tested.

Example 1: Find equation of hyperbola with focus (5, 0) and e = 5/3. Solution: Here c = 5, e = 5/3, so a = c/e = 3. Then b² = c² - a² = 25 - 9 = 16, so x²/9 - y²/16 = 1.

Example 2: Find latus rectum length of 16x² - 9y² = 144. Solution: Divide by 144: x²/9 - y²/16 = 1. Here a² = 9, b² = 16. Length = 4b²/a = 4(16)/3 = 64/3 units.

🔴 Extended — Deep Study (3mo+)

Comprehensive coverage for students on a longer timeline.

Hyperbola — Comprehensive CUET Notes

Theoretical Foundation

A hyperbola is a conic section formed when a plane cuts both nappes of a double cone. Formally, it’s the locus of points P where **|PF₁ - PF₂| Content adapted based on your selected roadmap duration. Switch tiers using the pill selector above.