If the curves $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ $(a > b)$ and $x^2 - y^2 = c^2$ intersect at right angles,then:

Let $PQ$ be a focal chord of the parabola $y^{2}=4x$ such that it subtends an angle of $\frac{\pi}{2}$ at the point $(3, 0)$. Let the line segment $PQ$ be also a focal chord of the ellipse $E: \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1, a^{2}>b^{2}$. If $e$ is the eccentricity of the ellipse $E$,then the value of $\frac{1}{e^{2}}$ is equal to

The slopes of the common tangents to the parabola $(x - 1)^2 = 4(y - 2)$ and the ellipse $\frac{(x - 1)^2}{1} + \frac{(y - 2)^2}{2} = 1$ are $m_1$ and $m_2$. Then,$m_1^2 + m_2^2$ is equal to:

Let $e_1$ be the eccentricity of the hyperbola $\frac{x^2}{16}-\frac{y^2}{9}=1$ and $e_2$ be the eccentricity of the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1, a>b$,which passes through the foci of the hyperbola. If $e_1 e_2=1$,then the length of the chord of the ellipse parallel to the $x$-axis and passing through $(0,2)$ is:

If the foci of $\frac{x^{2}}{16}+\frac{y^{2}}{4}=1$ and $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{9}=1$ coincide,then the value of $a$ is

If $e_1$,$e_2$,and $e_3$ are eccentricities of the conics $y = x^2 - x + 3$,$\frac{x^2}{a^2} + \frac{y^2}{3a^4} = 1$,and $a^2x^2 - 3a^4y^2 = 1$ respectively,then which of the following is correct? (where $a > 1$)