# Hyperbola (S.N.Dey) |Part-2 | Ex-6

###### In the previous article , we have solved few VSA type questions of Hyperbola chapter of S.N.Dey mathematics, Class 11. In this article , we will solve few more questions of Hyperbola related problems.

9(i) If the length of latus rectum of a rectangular hyperbola is unit, find its equation.

Solution.

The length of the latus rectum is

For rectangular hyperbola,

So, from we get,

Hence, the equation of the rectangular hyperbola is given by

(ii) Find the co-ordinates of the foci of the rectangular hyperbola

Solution.

Comparing the given hyperbola with we get, Also, we know that the eccentricity of any rectangular hyperbola  is

The co-ordinates of the foci is given by

10. If the latus rectum and the transverse axis of a hyperbola are equal, show that it is a rectangular hyperbola.

Solution.

We know that the length of latus rectum of the hyperbola  is unit and the transverse axis is given by unit.

By question,

Hence, by we get,

So, the given hyperbola is a rectangular hyperbola.

11. Find the positions of the points with respect to the hyperbola

Solution(i)

We know that the point lies outside, on or inside the hyperbola according as

The given equation of the hyperbola can be written as

Now, for the point we get,

Hence, by and we can conclude that the point lies inside the given hyperbola.

Solution(ii)

We know that the point lies outside, on or inside the hyperbola according as

The given equation of the hyperbola can be written as

Now, for the point we get,

Hence, by and we can conclude that the point lies inside the given hyperbola.

Solution(iii)

We know that the point lies outside, on or inside the hyperbola according as

The given equation of the hyperbola can be written as

Now, for the point we get,

Hence, by and we can conclude that the point lies outside the given hyperbola.

12. Find the position of the point with respect to the hyperbola

Solution.

We know that the point lies outside, on or inside the hyperbola according as

The given equation of the hyperbola can be written as

Now, for the point we get,

Hence, by and we can conclude that the point lies inside the given hyperbola.

13. Show that the locus of the point of intersection of the lines and being a variable parameter, is a hyperbola.

Solution.

The given equations of straight lines are

From and we get,

Hence, by we can conclude that the locus of point of intersection of the given lines is a hyperbola.

14. Find the parametric co-ordinates of the point on the hyperbola

Solution.

The equation of the hyperbola can be written as

Comparing with we get,

So, we can say that the any point on the hyperbola can be written in parametric form as

By question,

Hence, the parametric co-ordinates of the given point can be written as