Question 68 of 70intermediate🔧 ApplyLong Answer4 marks

Find the area of the triangle formed by the lines joining the vertex of the parabola x2=12yx^2 = 12y to the ends of its latus rectum.

Correct Answer

The area of the triangle is 18 square units.

Exercise: Miscellaneous Exercise on Chapter 10 | Q: 6 | (Page 29)
For More Understanding

Explanation

This question tests understanding of parabola properties and coordinate geometry. Students must identify the standard form x2=4ayx^2 = 4ay, extract the value of 'aa', locate the focus, determine the latus rectum endpoints, and then calculate the area of the triangle formed with the vertex.

Solution Steps

  1. Step 1: Compare the given equation x2=12yx^2 = 12y with the standard form x2=4ayx^2 = 4ay. We get 4a=124a = 12, so a = 3.

  2. Step 2: The vertex of the parabola is at the origin (0, 0). The focus lies at (0, a) = (0, 3).

  3. Step 3: The latus rectum is perpendicular to the axis (yy-axis) and passes through the focus. Its length is 4a = 12.

  4. Step 4: To find the endpoints of the latus rectum, substitute y=3y = 3 in x2=12yx^2 = 12y. We get x2=36x^2 = 36, so x=±6x = \pm 6. Thus, the endpoints are (6, 3) and (-6, 3).

  5. Step 5: The triangle has vertices at (0,0)(0, 0), (6,3)(6, 3), and (−6,3)(-6, 3). Using the area formula:

    Area=12×base×height=12×12×3=18 square units.\text{Area} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 12 \times 3 = \textbf{18 square units}.
PreviousNext