Length of latusrectum of curve $xy = 7x + 5y$ is

Let the tangent drawn to the parabola $y ^{2}=24 x$ at the point $(\alpha, \beta)$ is perpendicular to the line $2 x$ $+2 y=5$. Then the normal to the hyperbola $\frac{x^{2}}{\alpha^{2}}-\frac{y^{2}}{\beta^{2}}=1$ at the point $(\alpha+4, \beta+4)$ does $NOT$ pass through the point.

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

The equation of the tangent to the conic ${x^2} - {y^2} - 8x + 2y + 11 = 0$ at $(2, 1)$ is

Let $A$ be a point on the $x$-axis. Common tangents are drawn from $A$ to the curves $x^2+y^2=8$ and $y^2= 16x.$ If one of these tangents touches the two curves at $Q$ and $R$, then $( QR )^2$ is equal to

If the straight line $x\cos \alpha + y\sin \alpha = p$ be a tangent to the hyperbola $\frac{{{x^2}}}{{{a^2}}} - \frac{{{y^2}}}{{{b^2}}} = 1$, then