Question 16 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: y² = 12x

Correct Answer

The given equation of the parabola is y2=12xy^2 = 12x. Since the equation involves y2y^2, the axis of the parabola is the x-axis. The coefficient of xx is positive, so the parabola opens to the right.

Comparing y2=12xy^2 = 12x with the standard form y2=4axy^2 = 4ax, we find that 4a=124a = 12, which gives a=3a = 3. Therefore, the coordinates of the focus are (a,0)=(3,0)(a, 0) = (3, 0).

The equation of the directrix is x=−ax = -a, which is x=−3x = -3. The length of the latus rectum is 4a4a, which equals 4×3=124 \times 3 = 12.

Exercise: EXERCISE 10.2 | Q: 1 | (Page 11)
For More Understanding

Explanation

The solution follows the method demonstrated in Example 5 of the provided context. The standard equation y2=4axy^2 = 4ax is used to determine the value of 'a' by equating 4a4a to the coefficient of xx in the given equation. Once 'a' is found, the properties of the parabola (focus, directrix, latus rectum) are derived using the definitions provided in the text: focus at (a,0)(a,0), directrix at x=−ax=-a, and latus rectum length as 4a4a.

Solution Steps

  1. Step 1: Compare the given equation y2=12xy^2 = 12x with the standard form y2=4axy^2 = 4ax.

  2. Step 2: Solve for 'a' by equating 4a=124a = 12, resulting in a=3a = 3.

  3. Step 3: Identify the coordinates of the focus as (a,0)=(3,0)(a, 0) = (3, 0).

  4. Step 4: Identify the axis of the parabola as the x-axis (since y2y^2 is the squared term).

  5. Step 5: Determine the equation of the directrix as x=−a=−3x = -a = -3.

  6. Step 6: Calculate the length of the latus rectum using the formula 4a=4(3)=124a = 4(3) = 12.

PreviousNext