Question 22 of 70beginner🔧 ApplyShort Answer2 marks

In each of the Exercises 7 to 12, find the equation of the parabola that satisfies the given conditions: Focus (6,0); directrix x = -6

Correct Answer

Since the focus (6,0) lies on the x-axis, the axis of the parabola is the x-axis. The directrix is x = -6, which is a vertical line to the left of the origin. Therefore, the parabola opens to the right and its equation is of the form y² = 4ax.

Here, the focus is at (a, 0) and the directrix is x = -a, so comparing with given values, we get a = 6. Substituting this value, the required equation is y² = 4×64 \times 6 × x = 24x.

Exercise: EXERCISE 10.2 | Q: 7 | (Page 11)
For More Understanding

Explanation

The textbook derivation shows that for a parabola with focus (a,0) and directrix x = -a, the standard equation is y² = 4ax. Example 6 demonstrates the same method for focus (2,0) and directrix x = -2, yielding y² = 8x. Applying this to the given values, a = 6, so the equation is y² = 24x.

Solution Steps

  1. Step 1: Identify the axis: The focus (6,0) lies on the x-axis, so the axis of the parabola is the x-axis.

  2. Step 2: Determine the form: Since the directrix is x = -6 (vertical line left of origin) and focus is (6,0), the parabola opens to the right. Hence, its equation is of the form y² = 4ax.

  3. Step 3: Find the value of a: The distance from the vertex to the focus is a. Here, focus is (6,0) and directrix x = -6, so a = 6.

  4. Step 4: Write the equation: Substitute a = 6 into y² = 4ax to get y² = 24x.

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