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Ellipse
🟢 Lite — Quick Review (1h–1d)
Rapid summary for last-minute revision.
Ellipse — Key Facts for CUET • Standard equation: x²/a² + y²/b² = 1 (horizontal major axis if a > b, vertical if b > a) • Relationship: c² = a² - b², where c is the focal distance from center • Eccentricity: e = c/a (always between 0 and 1 for an ellipse) • Foci are at (±c, 0) for horizontal major axis, (0, ±c) for vertical major axis • Latus rectum length = 2b²/a—the chord through focus perpendicular to major axis • Directrices: x = ±a/e for horizontal axis, y = ±a/e for vertical axis ⚡ Exam tip: Remember—the sum of distances from any point on the ellipse to the two foci equals 2a
🟡 Standard — Regular Study (2d–2mo)
Standard content for students with a few days to months.
Ellipse — CUET Study Guide
An ellipse is the locus of points where the sum of distances from any point on the curve to two fixed points (foci) is constant and equals 2a, where a is the semi-major axis. The standard equation of an ellipse centered at origin with horizontal major axis is x²/a² + y²/b² = 1, where a > b. The semi-major axis length is a, semi-minor axis length is b. The relationship between a, b, and c (distance from center to each focus) is c² = a² - b². Eccentricity e = c/a measures the ellipse’s elongation; e = 0 is a circle, e → 1 is highly elongated.
The endpoints of the major axis are the vertices at (±a, 0), and the endpoints of the minor axis are (0, ±b). The latus rectum is the chord through a focus perpendicular to the major axis; its length is 2b²/a. Equations of directrices are x = ±a/e for horizontal major axis, helping define the ellipse using the focus-directrix property (distance to focus = e × distance to directrix).
Common CUET patterns: find equation given foci and sum of distances; find focus, eccentricity, and lengths given equation; convert between general and standard forms.
Example 1: Find equation of ellipse with foci (±4, 0) and major axis length 10. Solution: 2a = 10 → a = 5. c = 4 → b² = a² - c² = 25 - 16 = 9 → b = 3. Equation: x²/25 + y²/9 = 1.
Example 2: Find eccentricity and latus rectum of x²/36 + y²/16 = 1. Here a² = 36, b² = 16 → a = 6, b = 4. c² = 36 - 16 = 20 → c = √20 = 2√5. e = c/a = (2√5)/6 = √5/3. Latus rectum = 2b²/a = 2(16)/6 = 16/3.
🔴 Extended — Deep Study (3mo+)
Comprehensive coverage for students on a longer timeline.
Ellipse — Comprehensive CUET Notes
Advanced ellipse theory requires understanding auxiliary circles, director circles, and parametric representations. The auxiliary circle is the circle with radius a (for horizontal major axis)—its equation is x² + y² = a². The director circle is the locus of the intersection of perpendicular tangents to the ellipse; its equation is x² + y² = a² + b². When the ellipse is shifted with center (h, k), replace x with (x-h) and y with (y-k) in the standard equation.
Parametric equations: for x²/a² + y²/b² = 1, any point can be expressed as (a cosθ, b sinθ), where θ is the eccentric anomaly. The tangent at point (a cosθ₁, b sinθ₁) has equation (x cosθ₁)/a + (y sinθ₁)/b = 1. The normal at this point has equation a x secθ₁ - b y cosecθ₁ = (a² - b²). These forms frequently appear in advanced CUET problems involving chord equations, locus of intersection, and normals.
Cross-topic connections: Ellipses appear in planetary orbits (Kepler’s first law), architectural acoustics (whispering galleries), and engineering (optical reflectors). Connection with conic sections: ellipses, hyperbolas, and parabolas are all conics with different eccentricities. In coordinate geometry, transformation of axes can simplify ellipse equations. Challenge problems may ask for the equation of the chord of contact, chord bisected at a given point, or prove properties of conjugate diameters. Practice deriving equations of tangents in slope form: y = mx ± √(a²m² + b²) is tangent when m²a² > b².