Question

The area bounded by the parabola \( y^2 = 36x \) and its latus rectum is

• A. 216 sq units
• B. 108 sq units
• C. 27 sq units
• D. 54 sq units

Hint:

For parabola area with latus rectum, use symmetry and integrate with respect to \(y\) for easier limits.

Answer 1

The Correct Option is A

Step 1: Compare with standard parabola.
Given:
\[ y^2 = 36x \]
Standard form:
\[ y^2 = 4ax \Rightarrow 4a = 36 \Rightarrow a = 9 \]

Step 2: Find latus rectum.
For parabola \( y^2 = 4ax \), latus rectum is:
\[ x = a \]
So,
\[ x = 9 \]

Step 3: Find limits of integration.
At latus rectum:
\[ y^2 = 36(9) = 324 \Rightarrow y = \pm 18 \]

Step 4: Write area expression.
Area between parabola and latus rectum:
\[ A = \int_{-18}^{18} (9 - \frac{y^2}{36}) dy \]

Step 5: Use symmetry.
\[ A = 2 \int_{0}^{18} \left(9 - \frac{y^2}{36}\right) dy \]

Step 6: Integrate.
\[ A = 2\left[9y - \frac{y^3}{108}\right]_{0}^{18} \]
\[ = 2\left[162 - \frac{5832}{108}\right] \]
\[ = 2(162 - 54) \]
\[ = 2(108) = 216 \]

Step 7: Final conclusion.
Thus, the required area is 216 sq units.
Final Answer:
\[ \boxed{216} \]