Two Dimensional Analytical Geometry-II

y² = 12x

Step 1: A parabola with vertex at origin, axis along x-axis, opening to the right has the standard form y² = 4ax. Step 2: Here, a = 3. So 4a = 4 × 3 = 12. Step 3: The equation is y² = 12x. Step 4: The parabola opens to the right because the coefficient of x is positive. Final Answer: y² = 12x. Common mistake: Students confuse y² = 4ax (axis along x) with x² = 4ay (axis along y).

(2, 0)

Step 1: The standard form of a parabola is y² = 4ax, and its focus is at (a, 0). Step 2: Comparing y² = 8x with y² = 4ax: 4a = 8, so a = 2. Step 3: The focus is at (a, 0) = (2, 0). Final Answer: Focus = (2, 0). Common mistake: Students sometimes write the focus as (8, 0) by using the coefficient directly without dividing by 4, or place it on the y-axis.

16

Step 1: The standard form of the parabola is y² = 4ax. The length of the latus rectum is 4a. Step 2: Comparing y² = 16x with y² = 4ax: 4a = 16. Step 3: So the length of the latus rectum = 4a = 16. Final Answer: Latus rectum = 16 units. Common mistake: Students sometimes divide by 4 and write a = 4 as the answer, forgetting to state the latus rectum is 4a (which equals 16, not 4).

10

Step 1: The standard form of an ellipse is x²/a² + y²/b² = 1 with a > b. The major axis has length 2a. Step 2: Comparing with x²/25 + y²/9 = 1: a² = 25, so a = 5; b² = 9, so b = 3. Step 3: Since a > b (5 > 3), the major axis is along the x-axis. Step 4: Length of major axis = 2a = 2 × 5 = 10. Final Answer: Major axis length = 10 units. Common mistake: Students often write 5 (which is just a, the semi-major axis) instead of 10 (which is 2a, the full major axis).