The eccentricity of curve ${x^2} - {y^2} = 1$ is
$\frac{1}{2}$
$\frac{1}{{\sqrt 2 }}$
$2$
$\sqrt 2 $
(d) Since it is a rectangular hyperbola,
therefore eccentricity $e = \sqrt 2 $.
If the eccentricity of the hyperbola $x^2 – y^2 \sec^2 \alpha = 5$ is $\sqrt 3 $ times the eccentricity of the ellipse $x^2 \sec^2 \alpha + y^2 = 25, $ then a value of $\alpha$ is :
The locus of the foot of the perpendicular from the centre of the hyperbola $xy = c^2$ on a variable tangent is :
The combined equation of the asymptotes of the hyperbola $2{x^2} + 5xy + 2{y^2} + 4x + 5y = 0$
For hyperbola $\frac{{{x^2}}}{{{{\cos }^2}\alpha }} – \frac{{{y^2}}}{{{{\sin }^2}\alpha }} = 1$ which of the following remain constant if $\alpha$ varies
Let $a$ and $b$ be positive real numbers such that $a > 1$ and $b < a$. Let $P$ be a point in the first quadrant that lies on the hyperbola $\frac{ x ^2}{ a ^2}-\frac{ y ^2}{ b ^2}=1$. Suppose the tangent to the hyperbola at $P$ passes through the point $(1,0)$, and suppose the normal to the hyperbola at $P$ cuts off equal intercepts on the coordinate axes. Let $\Delta$ denote the area of the triangle formed by the tangent at $P$, the normal at $P$ and the $x$-axis. If $e$ denotes the eccentricity of the hyperbola, then which of the following statements is/are $TRUE$?
$(A)$ $1 < e < \sqrt{2}$
$(B)$ $\sqrt{2} < e < 2$
$(C)$ $\Delta=a^4$
$(D)$ $\Delta=b^4$
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