Dr. SK Goyal Solutions for Chapter: Hyperbola, Exercise 5: EXERCISE ON LEVEL-II

Find the range of parameter ' $a$ ' for which a unique circle will pass through the point of intersection of the rectangular hyperbola $x^2-y^2=a^2$  and the parabola $y=x^2$.

If the normals at $\left(x_i, y_i\right), i=1,2,3,4$ to the rectangular hyperbola $xy=2$ meet at the point $\left(3,4\right)$ , then $\frac{\sqrt{2}\sum t_1{t}_2{t}_3}{t_1{t}_2{t}_3{t}_4}$-

The normals at three points$P, Q, R$ on a rectangular hyperbola intersect at a point T on the curve. Prove that the centre of the hyperbola is the centroid of $∆PQR.$

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 ${\mathrm{a}}^2{\mathrm{x}}^2 - {\mathrm{b}}^2{\mathrm{y}}^2 = {\left({\mathrm{a}}^2+{\mathrm{b}}^2\right)}^2$.

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.

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

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.

$P$ 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 $\frac{{\left(2x-a\right)}^2}{a^2}-\frac{4y^2}{b^2}=1$: