Question:
What is the length of the latus rectum of the ellipse \(25x^2 + 16y^2 = 400\)?
दीर्घवृत्त \(25x^2 + 16y^2 = 400\) के नाभिलंब (latus rectum) की लंबाई क्या है?
Options:
- 25/2
- 25/4
- 16/5
- 32/5
- 25/2
- 25/4
- 16/5
- 32/5
Correct Answer: d)
Explanation:
Dividing \(25x^2 + 16y^2 = 400\) by 400:
\[\frac{x^2}{16} + \frac{y^2}{25} = 1\]
Here \(a^2 = 16\) and \(b^2 = 25\), so \(b = 5 > a = 4\).
Since \(b > a\), the major axis lies along the y-axis.
The length of the latus rectum is given by:
\[L = \frac{2a^2}{b} = \frac{2(16)}{5} = \frac{32}{5}\]
Hence, the correct option is (d).
समीकरण को मानक रूप में लिखने पर:
\[\frac{x^2}{16} + \frac{y^2}{25} = 1\]
यहाँ \(a^2 = 16\) तथा \(b^2 = 25\), जहाँ \(b = 5 > a = 4\)।
दीर्घ अक्ष y-अक्ष पर है, अतः नाभिलंब की लंबाई:
\[L = \frac{2a^2}{b} = \frac{2(16)}{5} = \frac{32}{5}\]
अतः सही विकल्प (d) है।