Hyperbola practice · Mixed difficulty

If a hyperbola passing through the origin has $3 x+4 y-1=0$ and $4 x-3 y-6=0$ as its assumptions, then the equation of the transverse axis of the hyperbola is

The locus of the equation ${x^2} - {y^2} = 0,$ is

If a diameter of a hyperbola meets the hyperbola in real points then

If a variable straight line $x \cos \alpha+y \sin \alpha=p$ which is a chord of hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1(b>a)$ subtends a right angle at the centre of the hyperbola, then it always touches a fixed circle whose radius, is

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

The equation of the chord of the hyperbola $25 x^2-16 y^2=400$ that is bisected at point $(5,3)$ is:

If the tangent at the point $P(a \sec \theta, b \tan \theta)$ to the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ passes through the point where a directrix of the hyperbola meets the positive side of the transverse axis, then $\theta$ is equal to
Where $e$ is the eccentricity of the hyperbola.

If tangent to any member of family of hyperbolas $x y=4 \sin ^2 \theta, \theta \in(0,2 \pi)-\{\pi \}$ is not a normal to any member of family of circles $x^2+y^2-2 x-2 y+\mu=0$, where $\mu$ is any real parameter, then $\theta$ belongs to

The equation of the diameter which is conjugate to $y$ $=3 x$ w.r.t. the hyperbola $\frac{x^2}{4}-\frac{y^2}{9}=1$, is