Question 21 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 = -9y

Correct Answer

The given equation is x² = -9y.

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

Step 2: Compare with standard form Comparing x² = -9y with x² = -4ay, we get: -4a = -9 Therefore, a = 9/4

Step 3: Find the focus For a parabola of the form x² = -4ay, the focus lies at (0, -a). Therefore, focus = (0, -9/4)

Step 4: Find the equation of directrix The equation of directrix is y = a. Therefore, directrix: y = 9/4

Step 5: Find the length of latus rectum Length of latus rectum = 4a = 4 × (9/4) = 9

Exercise: EXERCISE 10.2 | Q: 6 | (Page 11)
For More Understanding

Explanation

This question tests understanding of the standard form x² = -4ay for a downward-opening parabola. The key steps are: (1) recognizing that x² indicates the axis is the y-axis, (2) identifying that the negative coefficient means the parabola opens downward, (3) comparing with standard form to find 'a', and (4) applying the formulas for focus, directrix, and latus rectum. The focus is at (0, -a), the directrix is y = a, and the latus rectum length is 4a.

Solution Steps

  1. Step 1: Identify axis as y-axis (equation involves x²)

  2. Step 2: Compare x² = -9y with x² = -4ay to get a = 9/4

  3. Step 3: Focus = (0, -a) = (0, -9/4)

  4. Step 4: Directrix: y = a = 9/4

  5. Step 5: Length of latus rectum = 4a = 9

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