Question
Download Solution PDFदीर्घवृत्त \(\rm \frac{x^2}{100} + \frac{y^2}{75} = 1\) के नाभिलंब की लम्बाई क्या है?
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFसंकल्पना:
एक दीर्घवृत्त का मानक समीकरण:\(\frac{{{\rm{\;}}{{\bf{x}}^2}}}{{{{\bf{a}}^2}}} + \frac{{{{\bf{y}}^2}}}{{{{\bf{b}}^2}}} = 1\) (a > b)
- फोकस के निर्देशांक = (± ae, 0)
- उत्केंद्रता (e) = \(\sqrt {1 - {\rm{\;}}\frac{{{{\rm{b}}^2}}}{{{{\rm{a}}^2}}}} \) ⇔ a2e2 = a2 – b2
- नाभिलंब की लम्बाई = \(\rm \frac{2b^2}{a}\)
गणना:
दिया गया है: \(\rm \frac{x^2}{100} + \frac{y^2}{75} = 1\)
एक दीर्घवृत्त के मानक समीकरण के साथ तुलना करने पर: \(\frac{{{\rm{\;}}{{\bf{x}}^2}}}{{{{\bf{a}}^2}}} + \frac{{{{\bf{y}}^2}}}{{{{\bf{b}}^2}}} = 1\)
इसलिए, a2 = 100 और b2 = 75
∴ a = 10
नाभिलंब की लम्बाई = \(\rm \frac{2b^2}{a}\)= \(\rm \frac{2 \times 75}{10} = 15\)
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