The locus of a point which moves such that the difference of its distances from two fixed points is always a constant is

The midpoint of the chord $4x - 3y = 5$ of the hyperbola $2x^2 - 3y^2 = 12$ is

Let $O$ be the origin, and $P$ and $Q$ be two points on the rectangular hyperbola $xy = 12$ such that the midpoint of the line segment $PQ$ is $(\frac{1}{2}, -\frac{1}{2})$. Then the area of the triangle $OPQ$ equals:

If the straight line $(a-1)x - by + 4 = 0$ is normal to the hyperbola $xy = 1$, then which of the following does not hold?

The equations of the asymptotes of a hyperbola are $x+y+3=0$ and $2x-y+1=0$. If $(1,-2)$ is a point on this hyperbola,find the equation of its conjugate hyperbola.

The equation of the tangent parallel to $y-x+5=0$ drawn to $\frac{x^{2}}{3}-\frac{y^{2}}{2}=1$ is