Lf e₁ , e₂ and e₃ are eccentricities of conics y = {x^2} - x + 3,,frac{{{x^2}}}{{{a^2}}}

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10-2. Parabola, Ellipse, Hyperbola

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lf $e_1$ , $e_2$ and $e_3$ are eccentricities of the conics $y = {x^2} - x + 3,\,\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{3{a^4}}} = 1$ and ${a^2}{x^2} - 3{a^4}{y^2} = 1$ respectively, then which of the following is correct ? (where $a > 1)$

A

$e_3 < e_1 < e_2$

B

$e_2 < e_1 < e_3$

C

$e_3 < e_2 < e_1$

D

$e_1 < e_2 < e_3$

Solution

$y=x^{2}-x+3 \Rightarrow$ Parabola $\Rightarrow e_{1}=1$

$\frac{x^{2}}{a^{2}}+\frac{y^{2}}{3 a^{4}}=1 \Rightarrow$ ellipse $\Rightarrow e_{2}<1$

$a^{2} x^{2}-3 a^{4} y^{2}=1 \quad \Rightarrow$ Hyperbola $\Rightarrow e_{3}>1$

${e_2} < {e_1} < {e_3}$

Standard 11

Mathematics

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