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Parabola
🟢 Lite — Quick Review (1h–1d)
Rapid summary for last-minute revision.
Parabola — Key Facts for CUET • Standard equation: y² = 4ax (opens right), x² = 4ay (opens upward)—vertex at origin, focus at (a, 0) or (0, a) • Latus rectum length = 4a—the chord through the focus perpendicular to the axis • Focus-directrix property: distance from any point P to focus = distance from P to directrix • Vertex is midway between focus and directrix • If the parabola is translated, vertex shifts to (h, k)—substitute (x-h) and (y-k) in standard form • Most tested: finding equation from focus and directrix, and finding focus/directrix from equation ⚡ Exam tip: Use the definition PF = PD to derive equations quickly—never memorize all shifted forms
🟡 Standard — Regular Study (2d–2mo)
Standard content for students with a few days to months.
Parabola — CUET Study Guide
A parabola is the locus of points equidistant from a fixed point (focus) and a fixed line (directrix). The standard form with vertex at origin depends on the axis of symmetry: y² = 4ax opens rightward, x² = 4ay opens upward. The parameter “a” determines the distance from vertex to focus and from vertex to directrix—each equals |a|. The focus lies at (a, 0) or (0, a), and the directrix is the line x = -a or y = -a respectively. Eccentricity of a parabola is always 1, a defining property.
The latus rectum is the chord passing through the focus perpendicular to the axis. Its length is 4a, and its endpoints can be found by substituting x = a in y² = 4ax (giving y = ±2a). When the vertex shifts to (h, k), replace x with (x-h) and y with (y-k) in the standard equation. Common CUET patterns include: given focus and directrix, find the equation; given equation, identify vertex, focus, axis, and latus rectum; find the equation of tangent in slope form and point form.
Example 1: Find the equation of parabola with focus (4, 0) and directrix x = -4. Since focus is at (4, 0), a = 4, directrix x = -4. Using PF = PD: √[(x-4)² + y²] = |x + 4|. Squaring and simplifying gives (x-4)² + y² = (x+4)² → y² = 16x. Answer: y² = 16x.
Example 2: Find focus, directrix, and latus rectum length of y² = 8x. Comparing with y² = 4ax, 4a = 8 → a = 2. Focus: (2, 0), Directrix: x = -2, Latus rectum: 4a = 8.
🔴 Extended — Deep Study (3mo+)
Comprehensive coverage for students on a longer timeline.
Parabola — Comprehensive CUET Notes
Advanced study of parabolas requires understanding rotated and shifted forms, equation transformations, and tangents with proofs. For a parabola with vertex (h, k) and axis parallel to x-axis: (y-k)² = 4a(x-h). The focal parameter (distance from vertex to focus) is |a|. When the axis is parallel to y-axis: (x-h)² = 4a(y-k). The general second-degree equation of a parabola is Ax² + Bxy + Cy² + Dx + Ey + F = 0, where B² - 4AC = 0 identifies the parabola.
Tangents are derived using the condition of tangency: for y² = 4ax, the equation of tangent at point (x₁, y₁) is yy₁ = 2a(x + x₁). In slope form, if slope = m, the tangent is y = mx + a/m (for m ≠ 0). The point of intersection of tangents at the ends of a focal chord lies on the directrix. A focal chord connects two points on the parabola whose distances to the focus sum in a special way.
Cross-topic connections: Parabolas intersect with physics (projectile motion trajectories are parabolic), coordinate geometry (conic sections), and calculus (optimization problems). The reflective property of parabolas states that rays parallel to the axis reflect through the focus—this is used in telescope and headlight design. CUET Advanced may ask: prove that the product of slopes of perpendicular tangents to a parabola equals -1, or find the locus of intersection of tangents at points whose parametric parameters multiply to a constant. Practice deriving equations of normal chords and understanding the properties of director circles.