यदि दो दीर्घवृत्त `(x^(2))/(a^(2)) + (y^(2))/(b^(2)) = 1 "तथा" (x^(2))/(a^(2)) + (4y^(2))/(b^(2)) = 1` की उत्केंद्रताएँ क्रमशः `e_(1) "और" e_(2)` हो तो सिद्ध कीजिए कि ` 3 = 4 e_(2)^(2) - e_(1)^(2)` .

Explanation

दीर्घवृत्त `(x^(2)) /(2) + (y^(2))/(b^(2)) = 1 `
इसकी उत्केंद्रता `e_(1)` है |
` therefore b^(2) = a^(2) (1 - e_(1)^(2)) rArr (b^(2))/(a^(2)) = 1 = e_(1)^(2)` …(1)
दीर्घवृत्त `(x^(2))/(a^(2)) + (4y^(2))/(b^(2)) = 1 rArr (x^(2))/(a^(2)) + (y^(2))/(b^(2)//4) = 1`
इसकी उत्केंद्रता `e_(2)` है
` therefore (b^(2))/(4) = a^(2) (1 - e_(2)^(2))`
`rArr (b^(2))/(4a^(2)) = 1 - e_(2)^(2) rArr (b^(2))/(a^(2)) = 4 - 4e_(2)^(2)` ...(2)
समीकरण (1) और (2) से
` 4 - 4 e_(2)^(2) = 1 - e_(1)^(2)`
` rArr 3 = 4e_(2)^(2) - e_(1)^(2)` .