Amit M Agarwal Solutions for Chapter: Hyperbola, Exercise 4: Target Exercises

The product of the perpendicular from any point on the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ to its asymptotes, is equal to

If $H\left(x,y\right)=0$ represents the equation of a hyperbola and $A\left(x,y\right)=0, C\left(x,y\right)=0$ are the equation of its asymptotes and the conjugate hyperbola respectively, then for any point $\left(\alpha ,\beta \right)$ in the plane; $H\left(\alpha ,\beta \right), A\left(\alpha ,\beta \right)$ and $C\left(\alpha ,\beta \right)$ are in

If four points are taken on a rectangular hyperbola such that the chord joining any two is perpendicular to the chord joining the other two and $\alpha ,\beta ,\gamma ,\delta$ are the inclinations to either asymptote of the straight line joining these points to the centre, then

Let $P$ be a point on the hyperbola $x^2-y^2=a^2$, where $a$ is a parameter such that $P$ is nearest to the line $y=2x$. Then, the locus of $P$ is

The coordinates of the point of intersection of two tangents to a rectangular hyperbola referred to its asymptote as axes, are

From any point $P\left(h,k\right)$, four normals can be drawn to the rectangular hyperbola $xy=c^2$ such that sum of the ordinates of the feet of the normals is equal to

From any point $P\left(h,k\right)$, four normals can be drawn to the rectangular hyperbola $xy=c^2$ such that product of the abscissae of the feet of the normals $=$ product of the ordinates of the feet of the normals is equal to

A point $P$ moves in such a way that the sum of the slopes of the normals drawn from it to the hyperbola $xy=4$ is equal to the sum of the ordinates of feet of the normals. The locus of $P$ is a parabola $x^2=4y$. Then, the least distance of this parabola from the circle $x^2-y^2-24x+128=0$ is