Step 1: For a parabola, any point (x,y) on the parabola is equidistant to the focus and directrix.
Step 2: Distance to focus (4,0): √[(x-4)² + (y-0)²]
Step 3: Distance to directrix x=-4: |x - (-4)| = x + 4
Step 4: Set them equal: √[(x-4)² + y²] = x + 4
Step 5: Square both sides: (x-4)² + y² = (x+4)²
Step 6: Expand: x² - 8x + 16 + y² = x² + 8x + 16
Step 7: Simplify: y² = 16x
Step 1: Standard form of parabola opening left: y² = -4ax
Step 2: Compare with given equation y² = -8x → 4a = 8 → a = 2
Step 3: Vertex is at origin (0,0)
Step 4: Focus is at (-a,0) = (-2,0)
Step 5: Directrix is x = a → x = 2
Step 6: Length of latus rectum = 4a = 8
Step 1: Endpoints of major axis (0,±5) → major axis is y-axis, a = 5
Step 2: Foci at (0,±c) → c = 4
Step 3: For ellipse: b² = a² - c² = 25 - 16 = 9 → b = 3
Step 4: Standard form when major axis is y-axis: x²/b² + y²/a² = 1
Step 1: Compare with standard form x²/a² + y²/b² = 1
Step 2: a² = 25 → a = 5, b² = 9 → b = 3
Step 3: Since a > b, major axis is x-axis
Step 4: Center is at (0,0)
Step 5: Vertices at (±a,0) → (±5,0)
Step 6: c² = a² - b² = 25 - 9 = 16 → c = 4
Step 7: Foci at (±c,0) → (±4,0)
Step 1: For hyperbola, foci are at (±c,0) where c = ae
Step 2: Given c = 2 and e = 3/2 → a = c/e = 2/(3/2) = 4/3
Step 3: For hyperbola, b² = a²(e² - 1) = (16/9)(9/4 - 1) = (16/9)(5/4) = 20/9
Step 4: Standard equation when transverse axis is x-axis: x²/a² - y²/b² = 1
Step 1: Latus rectum is the chord through focus perpendicular to transverse axis
Step 2: For hyperbola x²/a² - y²/b² = 1, foci are at (±c,0) where c² = a² + b²
Step 3: Consider focus at (c,0). The latus rectum passes through (c,±y)
Step 4: Substitute (c,y) in equation: c²/a² - y²/b² = 1
Step 5: y²/b² = c²/a² - 1 = (a² + b²)/a² - 1 = b²/a²
Step 6: So y² = b⁴/a² → y = ±b²/a
Step 7: Length of latus rectum = distance between (c,b²/a) and (c,-b²/a) = 2b²/a
Step 1: Standard form of parabola opening upward: x² = 4ay
Step 2: Compare with given equation x² = 24y → 4a = 24 → a = 6
Step 3: Vertex is at origin (0,0)
Step 4: Focus is at (0,a) = (0,6)
Step 5: Directrix is y = -a → y = -6
Step 6: Length of latus rectum = 4a = 24
Step 1: Complete the square for x terms: x² - 2x = (x² - 2x + 1) - 1 = (x-1)² - 1
Step 2: Rewrite equation: (x-1)² - 1 + 8y + 17 = 0 → (x-1)² + 8y + 16 = 0
Step 3: Simplify: (x-1)² = -8y - 16 → (x-1)² = -8(y + 2)
Step 4: Compare with standard form (x-h)² = -4a(y-k)
Step 5: Vertex at (h,k) = (1,-2), 4a = 8 → a = 2
Step 6: Focus at (h,k-a) = (1,-2-2) = (1,-4)
Step 7: Directrix is y = k + a → y = -2 + 2 → y = 0
Step 1: Standard form of ellipse: (x-h)²/b² + (y-k)²/a² = 1 (since 289 > 225)
Step 2: Center at (h,k) = (3,4)
Step 3: a² = 289 → a = 17, b² = 225 → b = 15
Step 4: c² = a² - b² = 289 - 225 = 64 → c = 8
Step 5: Vertices at (h,k±a) = (3,4±17) → (3,21) and (3,-13)
Step 6: Foci at (h,k±c) = (3,4±8) → (3,12) and (3,-4)