The eccentricity of the ellipse $4{x^2} + 9{y^2} + 8x + 36y + 4 = 0$ is

Find the equation for the ellipse that satisfies the given conditions: Ends of major axis $(0,\, \pm \sqrt{5})$ ends of minor axis $(±1,\,0)$

The value of $\lambda $, for which the line $2x - \frac{8}{3}\lambda y = - 3$ is a normal to the conic ${x^2} + \frac{{{y^2}}}{4} = 1$ is

If the tangent to the parabola $y^2 = x$ at a point $\left( {\alpha ,\beta } \right)\,,\,\left( {\beta  > 0} \right)$ is also a tangent to the ellipse, $x^2 + 2y^2 = 1$, then $a$ is equal to

Equation of the ellipse with eccentricity $\frac{1}{2}$ and foci at $( \pm 1,\;0)$ is

Let $P(a\sec \theta ,\;b\tan \theta )$ and $Q(a\sec \varphi ,\;b\tan \varphi )$, where $\theta + \phi = \frac{\pi }{2}$, be two points on the hyperbola $\frac{{{x^2}}}{{{a^2}}} - \frac{{{y^2}}}{{{b^2}}} = 1$. If $(h, k)$ is the point of intersection of the normals at $P$ and $Q$, then $k$ is equal to