First slide

Hyperbola in conic sections

Question

$If e_{1} and e_{2} are the eccentricities of the ellipse, \frac{x^{2}}{18} + \frac{y^{2}}{4} = 1 and the hyperbola, \frac{x^{2}}{9} - \frac{y^{2}}{4} = 1$
$respectively and (e_{1}, e_{2}) is a point on the ellipse, 15 x^{2} + 3 y^{2} = k, then k is equal to:$

Moderate

A

17
B

16
C

15
D

14

Solution

$\begin{array}{l} We have e_{1} = \sqrt{1 - \frac{4}{18}} = \sqrt{\frac{7}{9}} and e_{2} = \sqrt{1 + \frac{4}{9}} = \sqrt{\frac{13}{9}} \\ Given (e_{1}, e_{2}) lie on 15 x^{2} + 3 y^{2} = k \Rightarrow 15 e_{1}^{2} + 3 e_{2}^{2} = k \Rightarrow k = 15 (\frac{7}{9}) + 3 (\frac{13}{9}) = 16 \end{array}$

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