Conic Sections

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A conic section is the locus of a point P which moves so that its distance from a fixed point is always in a constant ratio to its perpendicular distance from a fixed line. • Focus: The fixed point (

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Standard equation: y² = 4ax Key properties: • Vertex: (0, 0) • Focus: (a, 0) • Directrix: x = -a • Axis: y = 0 (x-axis) • Latus rectum: 4a • Opens rightward if a > 0, leftward if a < 0 Derivation: U

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Step 1: Identify the form Since focus is on x-axis and directrix is vertical, use y² = 4ax Step 2: Find value of 'a' Focus is (a, 0) = (2, 0), so a = 2 Directrix is x = -a = -2 ✓ (matches given) Ste

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1. y² = 4ax (opens right, a > 0) Focus: (a,0), Directrix: x = -a 2. y² = -4ax (opens left, a > 0) Focus: (-a,0), Directrix: x = a 3. x² = 4ay (opens up, a > 0) Focus: (0,a), Directrix: y =

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Method: Use tangent formula yy₁ = 2a(x + x₁) Step 1: Identify parameters Parabola: y² = 12x = 4(3)x, so a = 3 Point: (x₁, y₁) = (3, 6) Step 2: Verify point lies on parabola 6² = 12(3) → 36 = 36 ✓ S

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Ellipse: Locus of point whose distance from fixed point bears constant ratio (e < 1) to distance from fixed line. Standard equation: x²/a² + y²/b² = 1 (a > b) Key parameters: • Semi-major axis: a •

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Step 1: Convert to standard form 4x² + 9y² = 36 Divide by 36: x²/9 + y²/4 = 1 Step 2: Identify a² and b² Comparing with x²/a² + y²/b² = 1: a² = 9, b² = 4 So a = 3, b = 2 (since a > b) Step 3: Find e

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Step 1: Use given information Foci: (±ae, 0) = (±4, 0) Eccentricity: e = 1/3 Step 2: Find 'a' ae = 4 and e = 1/3 a × 1/3 = 4 a = 12 Step 3: Find 'b' e² = 1 - b²/a² (1/3)² = 1 - b²/144 1/9 = 1 - b²/1