Question

The foci of the ellipse \[ \frac{x^2}{25}+\frac{y^2}{16}=1 \] and that of the hyperbola \[ \frac{x^2}{a^2}-\frac{y^2}{b^2}=1 \] are same. The greatest length of the transverse axis of the hyperbola such that the difference of the squares of their eccentricities is at least one is

• A. \(\dfrac{15\sqrt2}{\sqrt{17}}\)
• B. \(6\)
• C. \(\dfrac{30}{\sqrt{17}}\)
• D. \(\dfrac{15}{\sqrt{34}}\)

Hint:

If an ellipse and a hyperbola have common foci, then \[ \boxed{a^2+b^2=c^2} \] for the hyperbola, where \(c\) is the common focal distance. Use \[ \boxed{e_{\text{ellipse}}^2=\frac{c^2}{a^2},\qquad e_{\text{hyperbola}}^2=\frac{c^2}{a^2}} \] to form the required inequality.

Answer 1

The Correct Option is A