G Tewani Solutions for Chapter: Hyperbola, Exercise 1: DPP

A rectangular hyperbola of latus rectum$4$ units passes through $\left(0, 0\right)$ and has $\left(2, 0\right)$ as its one focus. The equation of locus of the other focus is

If the curves $x^2-y^2=4$ and $xy=\sqrt{5}$ intersect at points $A$ and $B$, then the possible number of points $C$ on the curve $x^2-y^2=4$, such that triangle $ABC$ is equilateral, is

The point $\left(3\tan \left(\theta +60°\right), 2\tan \left(\theta +30°\right)\right)$, where $\theta$ is the parameter, lies on the conic. Then its center is

If $P\mathit{(}\alpha \mathit{,}\mathit{ }\beta \mathit{)}\mathit{,}$ the point of intersection of the ellipse $\frac{x^2}{a^2}+\frac{y^2}{a^2\left(1-e^2\right)}=1$ and the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{a^2\left(E^2-1\right)}=\frac{1}{4}$ is equidistant from the foci of the two curves (all lying on the right of $y$-axis), then

A hyperbola having transverse axis of length $\frac{1}{2}$ unit is confocal with the ellipse $3x^2+4y^2=12$. Then

In the $xy$ plane, the path defined by the equation $\frac{1}{x^m}+\frac{1}{y^m}+\frac{k}{{\left(x+y\right)}^n}=0$ is

A point moves such that the sum of the squares of its distances from the two sides of length $a$ of a rectangle is twice the sum of the squares of its distances from the other two sides of length $b$. The locus of the point can be

The equation of a hyperbola with co-ordinate axes as principal axes, and the distances of one of its vertices from the foci are $3$ and $1$, can be