Question 1: The equation of the directrix of a hyperbola is Its focus is and eccentricity . Find the equation of the hyperbola.

Answer:

Let be the focus and be a point on the Hyperbola.

Draw perpendicular from on the directrics. Then, by definition

This is the required equation of the hyperbola.

Question 2: Find the equation of the hyperbola whose

(i) focus is directrix is and eccentricity

(ii) focus is directrix is and eccentricity

(iii) focus is directrix is and eccentricity

(iv) focus is directrix is and eccentricity

(v) focus is directrix is and eccentricity

(vi) focus is directrix is and eccentricity

Answer:

(i) Given focus is directrix is and eccentricity

Let be the focus and be a point on the Hyperbola.

Draw perpendicular from on the directrics. Then, by definition

This is the required equation of the hyperbola.

(ii) Given focus is directrix is and eccentricity

Let be the focus and be a point on the Hyperbola.

Draw perpendicular from on the directrics. Then, by definition

This is the required equation of the hyperbola.

(iii) Given focus is directrix is and eccentricity

Let be the focus and be a point on the Hyperbola.

Draw perpendicular from on the directrics. Then, by definition

This is the required equation of the hyperbola.

(iv) Given focus is directrix is and eccentricity

Let be the focus and be a point on the Hyperbola.

Draw perpendicular from on the directrics. Then, by definition

This is the required equation of the hyperbola.

(v) Given focus is directrix is and eccentricity

Let be the focus and be a point on the Hyperbola.

Draw perpendicular from on the directrics. Then, by definition

This is the required equation of the hyperbola.

(vi) Given focus is directrix is and eccentricity

Let be the focus and be a point on the Hyperbola.

Draw perpendicular from on the directrics. Then, by definition

This is the required equation of the hyperbola.

Question 3: Find the eccentricity, coordinates of the foci, equations of directrices and length of the latus-rectum of the hyperbola

(i)

(ii)

(iii)

(iv)

(v)

Answer:

(i) Given

(ii) Given

(iii) Given

(iv) Given

(v) Given

Question 4: Find the axes, eccentricity, latus-rectum and the coordinates of the foci of the hyperbola

Answer:

Given

Question 5: Find the center, eccentricity, foci and directrices of the hyperbola

(i)

(ii)

(iii)

Answer:

(i) Given can be simplified in the following way:

Shifting the origin at without rotating the coordinate axes and denoting the new coordinates with respect to the new axes and

Center: w.r.t the new axes

Therefore the coordinates of the center with respect to old axes are

Equation of directrix :

Hence the equations of the directrices w.r.t the old axes are and

(ii) Given can be simplified in the following way:

Shifting the origin at without rotating the coordinate axes and denoting the new coordinates with respect to the new axes and

Center: w.r.t the new axes

Therefore the coordinates of the center with respect to old axes are

Equation of directrix :

Hence the equations of the directrices w.r.t the old axes are

(iii) Given can be simplified in the following way:

Shifting the origin at without rotating the coordinate axes and denoting the new coordinates with respect to the new axes and

Center: w.r.t the new axes

Therefore the coordinates of the center with respect to old axes are

Equation of directrix :

Hence the equations of the directrices w.r.t the old axes are

Question 6: Find the equation of the hyperbola, referred to its principal axes as axes of coordinates, in

the following cases:

(i) the distance between the foci and eccentricity

(ii) conjugate axis is and the distance between foci

(iii) conjugate axis is and passes through the point

Answer:

(i) Given the distance between the foci and eccentricity

The distance between the foci is

Therefore the equation of the hyperbola is given as

(ii) Given conjugate axis is and the distance between foci

The distance between the foci is

Length of conjugate axis,

Therefore the equation of the hyperbola is given as

(iii) Given conjugate axis is and passes through the point

Length of the conjugate axis

It passes through

Therefore the equation of the hyperbola is given as

Question 7: Find the equation of the hyperbola whose

(i) foci are and and eccentricity is

(ii) vertices are and and focus is

(iii) foci are and and eccentricity is

(iv) vertices are at and foci are

(v) vertices are at and one of the directrices is

(vi) foci at and eccentricity is

Answer:

(i) Given foci are and and eccentricity is

The center of the hyperbola is the mid point of the line joining the two foci.

Let and be the length of transverse and conjugate axes and let be the eccentricity

Then, the equation of the hyperbola is

Now, the distance between the two foci

This is the equation of the required hyperbola.

(ii) Given vertices are and and focus is

The center of the hyperbola is the mid point of the line joining the two foci.

Let and be the length of transverse and conjugate axes and let be the eccentricity

Then, the equation of the hyperbola is

Now, the distance between the two vertices

The distance between the focus and vertex

This is the equation of the required hyperbola.

(iii) Given foci are and and eccentricity is

The center of the hyperbola is the mid point of the line joining the two foci.

Let and be the length of transverse and conjugate axes and let be the eccentricity

Then, the equation of the hyperbola is

Now, the distance between the two foci

This is the equation of the required hyperbola.

(iv) Given vertices are at and foci are

Since the vertices are on y-axis, the equation of the required hyperbola is

The coordinates of its vertices and foci are and respectively.

Therefore and

Now,

We know,

This is the equation of the required hyperbola.

(v) Given vertices are at and one of the directrices is

The vertices of the hyperbola is

Therefore and

Given

We know

This is the equation of the required hyperbola.

(vi) Given foci at and eccentricity is

The foci of the hyperbola is

Therefore

We know

This is the equation of the required hyperbola.

Question 8: Find the eccentricity of the hyperbola, the length of whose conjugate axis is of the length of transverse axis.

Answer:

Let and be the transverse and conjugate axes and e be the eccentricity. Then,

Question 9: Find the equation of the hyperbola whose

(i) focus is at vertex at and center at

(ii) focus is at center at and

Answer:

(i) Given focus is at vertex at and center at

Let be the coordinate of the second vertex.

We know that the vertex of the hyperbola is the mid point of the line joining the two vertices.

Therefore the coordinates of the second vertex is

Let and be the length of transverse and conjugate axes and let be the eccentricity. The the equation of the hyperbola is

Now, the distance between the two vertices

Now, the distance between the vertex and focus is

Now,

Substituting and in equation i) we get

This is the equation of the required hyperbola.

(ii) Given focus is at center at and

Let be the coordinate of the second focus of the hyperbola.

We know that the vertex of the hyperbola is the mid point of the line joining the two foci.

Therefore the coordinates of the second focus is

Let and be the length of transverse and conjugate axes and let be the eccentricity. The the equation of the hyperbola is

Now, the distance between the two vertices

Given

Now,

Substituting and in equation i) we get

This is the equation of the required hyperbola.

Question 10: If is any point on the hyperbola whose axis are equal, prove that

Answer:

If the axes of the hyperbola are equal, then

The, the equation of hyperbola becomes

Thus the center and the focus are given by and respectively

Let be any point on hyperbola. Therefore it will satisfy the equation. We get,

Now,

Question 11: In each of the following find the equations of the hyperbola satisfying the given conditions:

(i) vertices foci

(ii) vertices foci

(iii) vertices foci

(iv) foci transverse axis

(v) foci conjugate axis

(vi) foci the latus-rectum

(vii) foci the latus-rectum

(viii) vertices

(ix) foci passing through

(x) foci latus-rectum

Answer:

(i) Given vertices foci

The coordinates of its vertices and foci are and respectively.

Substituting and in equation i) we get the equation of the required hyperbola

This is the equation of the required hyperbola.

(ii) Given vertices foci

Since the vertices lie on y-axis, so let the equation of the required hyperbola be

The coordinates of its vertices and foci are and respectively.

Substituting and in equation i) we get the equation of the required hyperbola

This is the equation of the required hyperbola.

(iii) Given vertices foci

Since the vertices lie on y-axis, so let the equation of the required hyperbola be

The coordinates of its vertices and foci are and respectively.

Substituting and in equation i) we get the equation of the required hyperbola

This is the equation of the required hyperbola.

(iv) Given foci transverse axis

The coordinates of its vertices and foci are and respectively.

Length of transverse axis

Substituting and in equation i) we get the equation of the required hyperbola

This is the equation of the required hyperbola.

(v) Given foci conjugate axis

Since the vertices lie on x-axis, so let the equation of the required hyperbola be

The length of the conjugate axis of the required hyperbola is 24

The coordinates of foci of the required hyperbola is (0, \pm be)

Substituting and in equation i) we get the equation of the required hyperbola

This is the equation of the required hyperbola.

(vi) Given foci the latus-rectum

Since the vertices lie on x-axis, so let the equation of the required hyperbola be

The length of the conjugate axis of the required hyperbola is

Now, the coordinates of foci of the required hyperbola is

Substituting and in equation i) we get the equation of the required hyperbola

This is the equation of the required hyperbola.

(vii) Given foci the latus-rectum

Since the vertices lie on x-axis, so let the equation of the required hyperbola be

The length of the latus-rectum of the required hyperbola is

Now, the coordinates of foci of the required hyperbola is

Substituting and in equation i) we get the equation of the required hyperbola

This is the equation of the required hyperbola.

(viii) Given vertices

Since the vertices lie on x-axis, so let the equation of the required hyperbola be

The length of the vertices of the required hyperbola is

Substituting and in equation i) we get the equation of the required hyperbola

This is the equation of the required hyperbola.

(ix) Given foci passing through

Since the vertices lie on y-axis, so let the equation of the required hyperbola be

It passes through Therefore

The coordinates of the foci for the required hyperbola are

Substituting in equation ii) we get

Substituting and in equation i) we get the equation of the required hyperbola

This is the equation of the required hyperbola.

(x) Given foci latus-rectum

Since the vertices lie on y-axis, so let the equation of the required hyperbola be

The length of the latus-rectum of the required hyperbola is

The coordinates of foci of the required hyperbola are

Considering positive value of

Therefore

Substituting and in equation i) we get the equation of the required hyperbola

This is the equation of the required hyperbola.

Question 12: If the distance between the foci of a hyperbola is and its eccentricity is , then obtain its equation.

Answer:

Eccentricity is

Distance between foci

Therefore the equation of hyperbola is

Question 13: Show that the set of all points such that the difference of their distances from and is always equal to represents a hyperbola.

Answer:

Let be a point of the set.

Distance of from

Distance of from

The difference between the distances

Squaring both sides we get

Squaring both sides, we get

This is the equation of the required hyperbola.