If the line $y = \sqrt{3}x$ cuts the curve $x^4 + ax^2y + bxy + cx + dy + 6 = 0$ at $A$,$B$,$C$,and $D$,then the value of $OA \cdot OB \cdot OC \cdot OD$ is,(where $O$ is the origin).

The equations of the common tangents to the two hyperbolas $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ and $\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1$ are:

Find the angle of intersection of the curves $y^{2}=4ax$ and $x^{2}=4by$.

If the tangent at the point $(1,2)$ on the ellipse $3x^2+4y^2=19$ is also a tangent to the parabola $y^2-kx=0$,then $k=$

Consider an ellipse $E$,a hyperbola $H$,and a parabola $P$ such that each curve has the focus at $(2, 3)$ and the corresponding directrix is $x + y - 10 = 0$. If $(\alpha, \alpha_1)$,$(\beta, \beta_1)$,and $(\gamma, \gamma_1)$ are the nearest vertices of the ellipse,hyperbola,and parabola to the given directrix respectively,then:

If the common tangents to the parabola $x^2 = 4y$ and the circle $x^2 + y^2 = 4$ intersect at the point $P$,then find the square of the slope of the line.