​ If e1 and e2 are respectively the eccentricities of the ellipse x^2/18 + y^2/4 = 1 and the hyperbola x^2 / 9 - y^2/4 = 1 then write the value of

Question

If e₁ and e₂ are respectively the eccentricities of the ellipse \(\frac{x^2}{18} + \frac{y^2}{4} = 1\) and the hyperbola \(\frac{x^2}{9} - \frac{y^2}{4} = 1, \) then write the value of 2e₁² + e₂².

Answer 1

Given: If e₁ and e₂ are respectively the eccentricities of the ellipse \(\frac{x^2}{18} + \frac{y^2}{4} = 1\) and the hyperbola \(\frac{x^2}{9} - \frac{y^2}{4} = 1, \) 

To find:  2e₁² + e₂².

Eccentricity(e) of hyperbola is given by,

For ellipse:

Eccentricity(e) of ellipse is given by,

Therefore,

Substituting values from (1) and (2) in 2e₁² + e₂² 

2e₁² + e₂²

Hence, value of 2e₁² + e₂² is 3