Question 19 of 70beginner🔧 ApplyShort Answer2 marks

In each of the following Exercises 1 to 6, find the coordinates of the focus, axis of the parabola, the equation of the directrix and the length of the latus rectum: x² = -16y

Correct Answer

The given equation is x² = -16y.

Step 1: Identify the form Since the equation involves x², the axis of symmetry is along the y-axis. The coefficient of y is negative, so the parabola opens downwards.

Step 2: Compare with standard form Comparing x² = -16y with x² = -4ay, we get: -4a = -16, so a = 4

Step 3: Find the required values

  • Coordinates of focus: Since the parabola opens downward, focus is at (0, -a) = (0, -4)
  • Axis of the parabola: The axis is the y-axis (line x = 0)
  • Equation of directrix: y = a, so y = 4
  • Length of latus rectum: 4a = 4(4) = 16 units
Exercise: EXERCISE 10.2 | Q: 4 | (Page 11)
For More Understanding

Explanation

This question tests understanding of the standard form x² = -4ay for a parabola opening downward. The context shows in Example 8 that x² = -4ay represents a parabola symmetric about the y-axis opening downward. From the latus rectum definition, the length equals 4a. Students must correctly identify a = 4 by comparing coefficients, then apply the formulas for focus (0, -a), directrix (y = a), and latus rectum length (4a).

Solution Steps

  1. Step 1: Identify that x² = -16y has axis along y-axis and opens downward

  2. Step 2: Compare with x² = -4ay to find a = 4

  3. Step 3: Apply formulas - Focus: (0, -4), Directrix: y = 4, Latus rectum: 16

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