If the line joining the foci of the hyperbola S1≡ (x2/a2)-(y2/b2)+1=0 does not subtend a right angle at any point on the hyperbola S2≡ (x2/4 a2)-(y2/ b2)=1, then number of integral values of 4e2 is/are ( e is eccentricity of S2=0 )

Question

If the line joining the foci of the hyperbola S1≡ (x2/a2)-(y2/b2)+1=0 does not subtend a right angle at any point on the hyperbola S2≡ (x2/4 a2)-(y2/ b2)=1, then number of integral values of 4e2 is/are ( e is eccentricity of S2=0 ) (A) 1 (B) 2 (C) 3 (D) 4

Tardigrade Admin · Accepted Answer

Circle having foci of the hyperbola S1 as the extremities of diameter should not intersect the hyperbola S2 at real points ⇒ a2+b2 < 4a2 ⇒ b2 < 3a2 ⇒ e2 < 7/4 ⇒ 4 < 4e2 < 7

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$2$

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$4$

Circle having foci of the hyperbola $S_{1}$ as the extremities of diameter should not intersect the hyperbola $S_{2}$ at real points
$\Rightarrow a^{2} + b^{2} < 4 a^{2}$
$\Rightarrow b^{2} < 3 a^{2}$
$\Rightarrow e^{2} < 7/4$
$\Rightarrow 4 < 4 e^{2} < 7$

Q. If the line joining the foci of the hyperbola S1​≡a2x2​−b2y2​+1=0 does not subtend a right angle at any point on the hyperbola S2​≡4a2x2​−b2y2​=1, then number of integral values of 4e2 is/are ( e is eccentricity of S2​=0 )

1

2

3

4

Circle having foci of the hyperbola S1​ as the extremities of diameter should not intersect the hyperbola S2​ at real points ⇒a2+b2<4a2 ⇒b2<3a2 ⇒e2<7/4 ⇒4<4e2<7

$1$

$2$

$3$

$4$

Circle having foci of the hyperbola $S_{1}$ as the extremities of diameter should not intersect the hyperbola $S_{2}$ at real points
$\Rightarrow a^{2} + b^{2} < 4 a^{2}$
$\Rightarrow b^{2} < 3 a^{2}$
$\Rightarrow e^{2} < 7/4$
$\Rightarrow 4 < 4 e^{2} < 7$