Let $P$ is a point on hyperbola $x^2 -y^2 = 4$ , which is at minimum distance from $(0,-1)$ then distance of $P$ from $x-$ axis is
$0$
$\frac{1}{2}$
$1$
$\sqrt 2 $
The values of parameter $'a'$ such that the line $\left( {{{\log }_2}\left( {1 + 5a - {a^2}} \right)} \right)x - 5y - \left( {{a^2} - 5} \right) = 0$ is a normal to the curve $xy = 1$ , may lie in the interval
A hyperbola passes through the foci of the ellipse $\frac{ x ^{2}}{25}+\frac{ y ^{2}}{16}=1$ and its transverse and conjugate axes coincide with major and minor axes of the ellipse, respectively. If the product of their eccentricities in one, then the equation of the hyperbola is ...... .
If area of quadrilateral formed by tangents drawn at ends of latus rectum of hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ is equal to square of distance between centre and one focus of hyperbola, then $e^3$ is ($e$ is eccentricity of hyperbola)
If the circle $x^2 + y^2 = a^2$ intersects the hyperbola $xy = c^2 $ in four points $ P(x_1, y_1), Q(x_2, y_2), R(x_3, y_3), S(x_4, y_4), $ then
The graph of the conic $ x^2 - (y - 1)^2 = 1$ has one tangent line with positive slope that passes through the origin. the point of tangency being $(a, b). $ Then Length of the latus rectum of the conic is