Question

The length of the latus rectum of \( 16x^2 + 25y^2 = 400 \) is:

• A. \( \frac{25}{2} \)
• B. \( \frac{25}{4} \)
• C. \( \frac{16}{2} \)
• D. \( \frac{32}{5} \)

Hint:

For an ellipse in standard form \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \), the length of the latus rectum is given by: \[ \frac{2b^2}{a}. \]

Answer 1

The Correct Option is D

Step 1: Convert the Given Equation to Standard Form The given equation is:
\[ 16x^2 + 25y^2 = 400. \] Dividing throughout by 400: \[ \frac{16x^2}{400} + \frac{25y^2}{400} = 1. \] \[ \frac{x^2}{25} + \frac{y^2}{16} = 1. \] This represents the standard form of an ellipse: \[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1, \] where: \[ a^2 = 25 \Rightarrow a = 5, \quad b^2 = 16 \Rightarrow b = 4. \] Step 2: Formula for Length of the Latus Rectum
The formula for the length of the latus rectum of an ellipse is: \[ \frac{2b^2}{a}. \] Substituting the values: \[ \frac{2(16)}{5} = \frac{32}{5}. \] Step 3: Conclusion
Thus, the correct answer is: \[ \mathbf{\frac{32}{5}}. \]