Conic Sections

Answer

Step 1: Use definition — distance from point to focus = distance to directrix. Let P(x, y) be any point on parabola. SP = PM ⇒ SP² = PM² (x - 3)² + (y - 0)² = (x + 3)² Expand: x² - 6x + 9 + y² = x² + …

Answer

Step 1: Use definition: SP = e × PM Let P(x, y) be any point. SP² = e² × PM² (x - 2)² + y² = (1/2)² × (x - 8)² Multiply both sides by 4: 4(x² - 4x + 4 + y²) = (x² - 16x + 64) 4x² - 16x + 16 + 4y² = x²…

Answer

Step 1: Compare with standard form y² = 4ax So, 4a = 16 ⇒ a = 4 Focus = (a, 0) = (4, 0) Latus rectum = 4a = 16 Answer: Focus (4, 0), Latus Rectum = 16…

Answer

Step 1: Write in standard form: Divide by 12: x²/4 + y²/3 = 1 So, a² = 4, b² = 3 Since a² > b², major axis is x-axis. e² = 1 - b²/a² = 1 - 3/4 = 1/4 e = √(1/4) = 1/2 Answer: e = 1/2…

Answer

Step 1: Use formula for tangent: yy₁ = 2a(x + x₁) Here, 4a = 8 ⇒ a = 2 Point (x₁, y₁) = (2, 4) So, y(4) = 2(2)(x + 2) 4y = 4(x + 2) 4y = 4x + 8 Divide by 4: y = x + 2 Answer: y = x + 2 or x - y + 2 = …

Answer

Step 1: For parabola y² = 4ax, 4a = 12 ⇒ a = 3 Slope of tangent = 2a/y₁ = 6/6 = 1 So slope of normal = -1/(slope of tangent) = -1 Equation: y - y₁ = m(x - x₁) y - 6 = -1(x - 3) y - 6 = -x + 3 x + y = …

Answer

Step 1: Foci at (±ae, 0) ⇒ ae = 4 e = 2/3 ⇒ a(2/3) = 4 ⇒ a = 6 Now, b² = a²(1 - e²) = 36(1 - 4/9) = 36(5/9) = 20 So equation: x²/36 + y²/20 = 1 Answer: x²/36 + y²/20 = 1…

Answer

Step 1: Compare with standard form x²/a² + y²/b² = 1 a² = 25, b² = 9 ⇒ a = 5, b = 3 Since a > b, major axis is x-axis. Length of latus rectum = 2b²/a = 2(9)/5 = 18/5 = 3.6 Answer: 3.6 units…