Rectangular Hyperbola

If the sum of the slopes of the normals from a point $P$ on hyperbola $xy=c^2$ is constant $k\left(k>0\right)$, then the locus of $P$ is

If the tangent and normal to a rectangular hyperbola cut off intercepts $x_1$ and $x_2$ on one axis and $y_1$ and $y_2$ on the other axis, then:

If a triangle is inscribed in a rectangular hyperbola, its orthocentre lies:

$P$ is a variable point on the hyperbola in the form $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$, whose vertex $A$ is $\left(a,0\right)$. Show that the locus of the mid point of $AP$ is $\frac{{\left(2x-a\right)}^2}{a^2}-\frac{4y^2}{b^2}=1$:

A rectangular hyperbola has double contact with a fixed central conic. If the chord of contact always passes through a fixed point. Prove that the locus of the centre of the hyperbola is a circle passing through the centre of the fixed conic.

Prove that the circles described on the four sides of a rhombus as diameters, pass through the point of intersection of its diagonals.

Show that, if a rectangular hyperbola cut a circle in four points, the centre of mean position of the four points is midway between the centres of the two curves.

A normal to the hyperbola $\frac{{\mathrm{x}}^2}{{\mathrm{a}}^2}-\frac{{\displaystyle {\mathrm{y}}^2}}{{\displaystyle {\mathrm{b}}^2}}=1$ meets the axes at $\mathrm{M} \text{and} \mathrm{N}$ and lines $\mathrm{MP}$ and $\mathrm{NP}$ are drawn perpendicular to axes meeting at $\mathrm{P}$. Prove that the locus of $\mathrm{P}$ is the hyperbola ${\mathrm{a}}^2{\mathrm{x}}^2 - {\mathrm{b}}^2{\mathrm{y}}^2 = {\left({\mathrm{a}}^2+{\mathrm{b}}^2\right)}^2$.

An ellipse has eccentricity $\frac{1}{2}$ and one focus at the point $P\left(\frac{1}{2},1\right).$ Its one directrix is the common tangent nearer to the point $P,$ to the circle $x^2+y^2=1$ and the hyperbola $x^2-y^2=1.$ The equation of the ellipse in the standard form is

The eccentricity of the conic represented by $x^2-y^2-4x+4y+16=0$ is

If the chords of the hyperbola $x^2-y^2=a^2$ touch the parabola $y^2=4ax$. Then, the locus of the middle points of these chords is

The area of triangle formed by the lines $x-y=0,$ $x+y=0$ and any tangent to the hyperbola $x^2-y^2=a^2,$ is