The transformed equation of $x^2-y^2+2 x+4 y=0$ when the origin is shifted to the point $(-1,2)$ is

If $e_1$ and $e_2$ are respectively the eccentricities of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ and its conjugate hyperbola, then the line $\frac{x}{2 e_1}+\frac{y}{2 e_2}=1$ touches the circle having centre at the origin, then its radius is

The locus of point of intersection of tangents at the ends of normal chord of the hyperbola $$x^2-y^2=a^2$$ is

If $$e_1$$ and $$e_2$$ are the eccentricities of the hyperbola $$16 x^2-9 y^2=1$$ and its conjugate respectively. Then, $$3 e_1=$$

If the normal to the rectangular hyperbola $$x^2-y^2=1$$ at the point $$P(\pi / 4)$$ meets the curve again at $$Q(\theta)$$, then $$\sec ^2 \theta+\tan \theta=$$

If the vertices and foci of a hyperbola are respectively $$( \pm 3,0)$$ and $$( \pm 4,0)$$, then the parametric equations of that hyperbola are

The value of $$\frac{1+\tan \mathrm{h} x}{1-\tan \mathrm{h} x}$$ is

Let origin be the centre, $$( \pm 3,0)$$ be the foci and $$\frac{3}{2}$$ be the eccentricity of a hyperbola. Then, the line $$2 x-y-1=0$$

The locus of a variable point whose chord of contact w.r.t. the hyperbola $$\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$$ subtends a right angle at the origin is

If one focus of a hyperbola is $$(3,0)$$, the equation of its directrix is $$4 x-3 y-3=0$$ and its eccentricity $$e=5 / 4$$, then the coordinates of its vertex is

The asymptotes of the hyperbola $$\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$$, with any tangent to the hyperbola form a triangle whose area is $$a^2 \tan (\alpha)$$. Then, its eccentricity equals